ESIGELEC IRSEEM Higher School of Engineering Entrance Exam Set

The core principle tested here is the understanding of signal-to-noise ratio (SNR) in the context of digital communication systems, a fundamental concept in electrical engineering and information theory, highly relevant to ESIGELEC IRSEEM’s curriculum in telecommunications and signal processing. The question assesses the ability to analyze how changes in signal power and noise power affect the overall clarity and reliability of a transmitted signal. The signal-to-noise ratio (SNR) is defined as the ratio of the power of a signal to the power of the background noise. Mathematically, it is often expressed in decibels (dB) as: \[ \text{SNR}_{\text{dB}} = 10 \log_{10} \left( \frac{P_{\text{signal}}}{P_{\text{noise}}} \right) \] where \( P_{\text{signal}} \) is the signal power and \( P_{\text{noise}} \) is the noise power. In the given scenario, the initial SNR is 20 dB. Let the initial signal power be \( P_{s1} \) and the initial noise power be \( P_{n1} \). \[ 20 = 10 \log_{10} \left( \frac{P_{s1}}{P_{n1}} \right) \] \[ 2 = \log_{10} \left( \frac{P_{s1}}{P_{n1}} \right) \] \[ \frac{P_{s1}}{P_{n1}} = 10^2 = 100 \] Now, the signal power is increased by a factor of 5, so the new signal power is \( P_{s2} = 5 P_{s1} \). The noise power is also increased by a factor of 2, so the new noise power is \( P_{n2} = 2 P_{n1} \). The new SNR is: \[ \text{SNR}_{\text{dB, new}} = 10 \log_{10} \left( \frac{P_{s2}}{P_{n2}} \right) \] \[ \text{SNR}_{\text{dB, new}} = 10 \log_{10} \left( \frac{5 P_{s1}}{2 P_{n1}} \right) \] \[ \text{SNR}_{\text{dB, new}} = 10 \log_{10} \left( \frac{5}{2} \times \frac{P_{s1}}{P_{n1}} \right) \] Substitute the ratio \( \frac{P_{s1}}{P_{n1}} = 100 \): \[ \text{SNR}_{\text{dB, new}} = 10 \log_{10} \left( \frac{5}{2} \times 100 \right) \] \[ \text{SNR}_{\text{dB, new}} = 10 \log_{10} (2.5 \times 100) \] \[ \text{SNR}_{\text{dB, new}} = 10 \log_{10} (250) \] To calculate \( \log_{10}(250) \): \( \log_{10}(250) = \log_{10}(100 \times 2.5) = \log_{10}(100) + \log_{10}(2.5) \) \( \log_{10}(100) = 2 \) \( \log_{10}(2.5) \approx 0.3979 \) So, \( \log_{10}(250) \approx 2 + 0.3979 = 2.3979 \) \[ \text{SNR}_{\text{dB, new}} \approx 10 \times 2.3979 \approx 23.979 \text{ dB} \] Rounding to one decimal place, the new SNR is approximately 24.0 dB. This question is crucial for understanding the practical implications of signal processing and communication system design, areas of significant focus at ESIGELEC IRSEEM. A higher SNR directly translates to improved data integrity and reduced error rates in digital transmissions, which is paramount for applications like wireless communication, embedded systems, and advanced sensor networks that are core to the school’s research and teaching. The ability to quantify and predict the impact of power fluctuations on signal quality is a foundational skill for any engineer working in these fields. Understanding the logarithmic nature of decibel scales and how multiplicative changes in power translate to additive changes in decibels is a key analytical skill.

The core principle being tested here is the understanding of electromagnetic wave propagation in a dielectric medium and its interaction with a conductive surface. When an electromagnetic wave encounters a boundary between two media, reflection and transmission occur based on the properties of the media. For a wave incident on a perfect conductor, the electric field component of the wave must be zero at the surface. This boundary condition dictates that the incident electric field and the reflected electric field must cancel each other out at the surface. Consequently, the reflected wave will have the same amplitude as the incident wave but will be phase-shifted by 180 degrees (or \(\pi\) radians) relative to the incident wave. This phase shift ensures that the total electric field at the boundary is zero. The magnetic field component, however, behaves differently. At the surface of a perfect conductor, the tangential component of the magnetic field is also zero. This leads to a phase shift of 180 degrees for the reflected magnetic field as well, ensuring the total tangential magnetic field is zero. The question probes the understanding of these fundamental boundary conditions and their implications for the phase of the reflected wave components, a concept crucial in fields like antenna theory, microwave engineering, and optical coatings, all relevant to ESIGELEC IRSEEM’s curriculum. The specific scenario of a plane wave incident normally on a perfect conductor is a foundational case study in electromagnetics.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning aliasing and sampling. Aliasing occurs when a signal is sampled at a rate lower than twice its highest frequency component (Nyquist rate). If a signal contains frequencies above \(f_{max}\), and it is sampled at a rate \(f_s\), then frequencies above \(f_s/2\) will appear as lower frequencies in the sampled signal. Specifically, a frequency \(f\) greater than \(f_s/2\) will be aliased to \(|f – k \cdot f_s|\) for some integer \(k\) such that \(|f – k \cdot f_s| \le f_s/2\). Consider a signal with components at 10 kHz and 25 kHz. If this signal is sampled at 40 kHz, the Nyquist frequency is \(f_s/2 = 40 \text{ kHz} / 2 = 20 \text{ kHz}\). The 10 kHz component is below the Nyquist frequency, so it will be represented correctly. The 25 kHz component is above the Nyquist frequency. To find its aliased frequency, we look for \(|25 \text{ kHz} – k \cdot 40 \text{ kHz}|\) such that the result is less than or equal to 20 kHz. For \(k=1\), \(|25 \text{ kHz} – 1 \cdot 40 \text{ kHz}| = |-15 \text{ kHz}| = 15 \text{ kHz}\). Since 15 kHz is less than or equal to 20 kHz, the 25 kHz component will be aliased to 15 kHz. Therefore, after sampling at 40 kHz, the signal will appear to have components at 10 kHz and 15 kHz. The presence of the aliased 15 kHz component, which was originally 25 kHz, means that the original 25 kHz frequency cannot be distinguished from the 15 kHz frequency in the sampled data. This phenomenon is a critical consideration in the design of data acquisition systems at institutions like ESIGELEC IRSEEM Higher School of Engineering, where accurate signal representation is paramount for research and development in fields like telecommunications and embedded systems. Understanding aliasing is crucial for selecting appropriate sampling rates and implementing anti-aliasing filters to preserve signal integrity and avoid misinterpretations of data, which directly impacts the validity of experimental results and the performance of engineered systems.

The question probes the understanding of the fundamental principles governing the operation of a synchronous generator, specifically focusing on the relationship between excitation current, terminal voltage, and power factor under varying load conditions. For a synchronous generator operating at a constant terminal voltage and frequency, an increase in excitation current (field current) leads to a rise in the internal generated electromotive force (EMF), denoted as \(E_a\). This \(E_a\) is the voltage generated within the armature windings due to the magnetic field produced by the field winding. The terminal voltage, \(V_t\), is related to \(E_a\) by the equation \(V_t = E_a – I_a Z_a\), where \(I_a\) is the armature current and \(Z_a\) is the synchronous impedance of the armature. When the excitation is increased beyond the level required for unity power factor operation at a given load, \(E_a\) becomes larger than \(V_t\) for a lagging power factor load, and \(E_a\) becomes smaller than \(V_t\) for a leading power factor load. To maintain a constant terminal voltage \(V_t\), the armature current \(I_a\) must adjust. If \(E_a\) increases while \(V_t\) is held constant, and the load is such that the power factor is lagging, the armature current \(I_a\) will decrease, and the power factor will tend towards unity or even leading. Conversely, if the load power factor is leading, increasing excitation further will cause \(E_a\) to become even more dominant relative to \(V_t\), leading to a larger \(I_a\) and a more leading power factor. The “V-curve” for a synchronous generator illustrates this: plotting armature current against field current at constant load power shows a U-shape. The minimum point of the V-curve corresponds to unity power factor operation. Operating to the left of the minimum (lower field current) results in a lagging power factor, while operating to the right (higher field current) results in a leading power factor. Therefore, increasing excitation beyond the unity power factor point for a given load will cause the generator to operate at a leading power factor, and if the terminal voltage is maintained constant, the armature current will adjust accordingly. The question asks about the effect of increasing excitation while maintaining constant terminal voltage and load power. For a fixed load power, the power factor is determined by the relationship between \(E_a\) and \(V_t\). Increasing excitation increases \(E_a\). If the generator was initially operating at unity power factor, increasing excitation will cause \(E_a\) to exceed \(V_t\) (considering the impedance drop), leading to a leading power factor. If the generator was already operating at a leading power factor, further increasing excitation will make the power factor even more leading. If it was operating at a lagging power factor, increasing excitation can bring it towards unity and then into a leading power factor. The most significant and direct consequence of increasing excitation beyond the unity power factor point, while maintaining constant terminal voltage and load power, is the shift towards a leading power factor. The armature current’s behavior is dependent on the initial power factor and the magnitude of the excitation increase, but the power factor shift is a direct consequence of the increased internal EMF relative to the terminal voltage.

The question probes the understanding of signal processing concepts, specifically focusing on the trade-offs in digital filter design. A causal FIR filter of order \(N\) has \(N+1\) coefficients. The question implies a scenario where a specific frequency response characteristic is desired, and the student must infer the most appropriate filter design strategy based on the constraints. A key principle in FIR filter design is the relationship between filter order, complexity, and performance. Higher-order filters generally offer sharper transitions between passbands and stopbands and better attenuation, but at the cost of increased computational complexity (more multiplications and additions per sample) and potentially higher latency. Conversely, lower-order filters are computationally less demanding but provide less precise frequency selectivity. The scenario presented, concerning the need for precise control over spectral characteristics in a high-frequency application at ESIGELEC IRSEEM Higher School of Engineering, suggests that a simple, low-order filter might not suffice. The desire for “fine-grained spectral shaping” points towards a filter that can approximate a complex frequency response with high fidelity. While IIR filters can achieve sharper transitions with lower orders, they introduce non-linear phase responses, which can be problematic in certain high-frequency applications where phase distortion is critical. Causal FIR filters, on the other hand, can be designed to have linear phase, making them suitable for applications where phase integrity is paramount. Designing a linear-phase FIR filter to meet stringent spectral requirements often necessitates a higher order compared to an IIR filter achieving a similar magnitude response. Therefore, to achieve the specified “fine-grained spectral shaping” and maintain phase linearity in a high-frequency context, a higher-order causal FIR filter is the most appropriate choice, despite its increased computational cost. The explanation does not involve a numerical calculation, but rather a conceptual understanding of filter design principles and their application in engineering contexts relevant to ESIGELEC IRSEEM Higher School of Engineering.

The question probes the understanding of the fundamental principles of electromagnetic induction and Lenz’s Law, particularly as applied to dynamic scenarios within electrical engineering contexts relevant to ESIGELEC IRSEEM Higher School of Engineering. Lenz’s Law states that the direction of induced current in a conductor will be such that it opposes the change in magnetic flux that produced it. In this scenario, a conducting loop is moving into a region of uniform magnetic field. As the loop enters the field, the magnetic flux through the loop increases. According to Lenz’s Law, an induced current will flow in the loop to create a magnetic field that opposes this increase. This opposing magnetic field will be directed opposite to the external field. To create an opposing magnetic field, the induced current must flow in a specific direction within the loop. If the external magnetic field is directed into the page, the induced magnetic field must be directed out of the page, which, by the right-hand rule, corresponds to a counter-clockwise current. Conversely, if the external field is out of the page, the induced field must be into the page, requiring a clockwise current. The key is that the induced current’s magnetic field *opposes* the *change* in flux. As the loop continues to move into the field, the flux continues to increase, and the induced current persists. Once the loop is fully within the uniform field, the flux is no longer changing, and thus no current is induced. As the loop exits the field, the flux decreases, and an induced current will flow to oppose this decrease, creating a magnetic field in the same direction as the external field. Therefore, induced current is present only during the periods of flux change, i.e., when the loop is entering or exiting the magnetic field. The magnitude of the induced current is proportional to the rate of change of magnetic flux, which in turn depends on the velocity of the loop and the rate at which the area within the field changes. The question requires understanding that induced current is a consequence of a changing magnetic flux, not merely the presence of a magnetic field. This concept is foundational for understanding generators, transformers, and various other electromagnetic devices studied at ESIGELEC IRSEEM Higher School of Engineering.

The question probes the understanding of a fundamental concept in digital signal processing, specifically related to the Nyquist-Shannon sampling theorem and its implications for aliasing. The scenario describes a continuous-time signal \(x(t)\) with a maximum frequency component of \(f_{max} = 15 \text{ kHz}\). According to the Nyquist-Shannon sampling theorem, to perfectly reconstruct a continuous-time signal from its discrete samples, the sampling frequency \(f_s\) must be at least twice the maximum frequency component of the signal. This minimum sampling frequency is known as the Nyquist rate, \(f_N = 2 \times f_{max}\). In this case, \(f_{max} = 15 \text{ kHz}\). Therefore, the Nyquist rate is \(f_N = 2 \times 15 \text{ kHz} = 30 \text{ kHz}\). If the signal is sampled at a frequency \(f_s\) less than the Nyquist rate, aliasing will occur. Aliasing is the phenomenon where higher frequencies in the original signal are misinterpreted as lower frequencies in the sampled signal, leading to distortion and loss of information. The question asks for the highest frequency component that can be unambiguously represented in the sampled signal. This highest unambiguous frequency is half of the sampling frequency, known as the folding frequency or Nyquist frequency, \(f_{Nyquist} = f_s / 2\). The scenario states that the signal is sampled at \(f_s = 25 \text{ kHz}\). Since \(f_s = 25 \text{ kHz} < 30 \text{ kHz}\) (the Nyquist rate), aliasing will occur. The highest frequency that can be unambiguously represented is \(f_s / 2 = 25 \text{ kHz} / 2 = 12.5 \text{ kHz}\). Any frequency component in the original signal above \(12.5 \text{ kHz}\) will be aliased into the frequency range of \(0\) to \(12.5 \text{ kHz}\). For instance, the \(15 \text{ kHz}\) component will appear as \(|15 \text{ kHz} – 25 \text{ kHz}| = |-10 \text{ kHz}| = 10 \text{ kHz}\) in the sampled signal. Therefore, the highest frequency that can be unambiguously represented without distortion due to aliasing is \(12.5 \text{ kHz}\). This concept is crucial in digital signal processing, a core area within ESIGELEC IRSEEM Higher School of Engineering's curriculum, ensuring that students understand the practical limitations and requirements for accurate digital representation of analog signals.

The question probes the understanding of a fundamental concept in digital signal processing, specifically related to the Nyquist-Shannon sampling theorem and its implications for aliasing. The scenario describes an analog signal \(x(t) = 5 \cos(200\pi t) + 2 \sin(600\pi t)\). The maximum frequency component in this signal is determined by the highest frequency term. The first term, \(5 \cos(200\pi t)\), has an angular frequency \(\omega_1 = 200\pi\) radians per second. The corresponding frequency \(f_1\) is given by \(\omega_1 = 2\pi f_1\), so \(f_1 = \frac{200\pi}{2\pi} = 100\) Hz. The second term, \(2 \sin(600\pi t)\), has an angular frequency \(\omega_2 = 600\pi\) radians per second. The corresponding frequency \(f_2\) is given by \(\omega_2 = 2\pi f_2\), so \(f_2 = \frac{600\pi}{2\pi} = 300\) Hz. Therefore, the maximum frequency component in the analog signal \(x(t)\) is \(f_{max} = 300\) Hz. According to the Nyquist-Shannon sampling theorem, to perfectly reconstruct an analog signal from its samples, the sampling frequency \(f_s\) must be at least twice the maximum frequency component of the signal. This minimum sampling frequency is known as the Nyquist rate, \(f_{Nyquist} = 2 f_{max}\). In this case, the Nyquist rate is \(2 \times 300 \text{ Hz} = 600\) Hz. The question states that the signal is sampled at a rate of \(f_s = 400\) Hz. Since \(f_s = 400\) Hz is less than the Nyquist rate of 600 Hz, aliasing will occur. Aliasing is the phenomenon where high-frequency components in the analog signal are incorrectly interpreted as lower frequencies in the sampled signal. The aliased frequency \(f_{alias}\) for a frequency \(f\) sampled at \(f_s\) is given by \(f_{alias} = |f – k f_s|\), where \(k\) is an integer chosen such that \(f_{alias}\) falls within the range \([0, f_s/2]\). Let’s consider the two frequency components: 1. For the 100 Hz component: Since \(100 \text{ Hz} < f_s/2 = 400/2 = 200 \text{ Hz}\), this component will not be aliased. Its sampled frequency remains 100 Hz. 2. For the 300 Hz component: Since \(300 \text{ Hz} > f_s/2 = 200 \text{ Hz}\), this component will be aliased. We need to find an integer \(k\) such that \(|300 – k \times 400|\) is minimized and within \([0, 200]\). – If \(k=1\), \(|300 – 1 \times 400| = |-100| = 100\) Hz. This is within the range \([0, 200]\). – If \(k=0\), \(|300 – 0 \times 400| = 300\) Hz, which is outside the range. – If \(k=2\), \(|300 – 2 \times 400| = |-500| = 500\) Hz, which is outside the range. Thus, the 300 Hz component will appear as 100 Hz in the sampled signal. The sampled signal will therefore contain components at 100 Hz (from the original 100 Hz component) and 100 Hz (from the aliased 300 Hz component). This means the sampled signal will appear as a single sinusoid at 100 Hz. The amplitude of the resulting 100 Hz component will be the sum of the amplitudes of the original 100 Hz component and the aliased 300 Hz component, which is \(5 + 2 = 7\). Therefore, the sampled signal will be equivalent to \(7 \cos(200\pi t)\). The question asks for the frequency of the resulting signal, which is 100 Hz. This understanding is crucial for engineers at ESIGELEC IRSEEM Higher School of Engineering Entrance Exam University, particularly in fields like telecommunications and embedded systems, where signal integrity and accurate data acquisition are paramount. Proper sampling is a cornerstone of digital signal processing, and recognizing the conditions for aliasing and its consequences is a fundamental skill.

The question assesses understanding of the fundamental principles of electromagnetic induction and Faraday’s Law, particularly in the context of a changing magnetic flux within a conductor. The scenario describes a metallic ring placed within a time-varying magnetic field. Faraday’s Law states that the induced electromotive force (EMF) in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. Mathematically, this is expressed as \( \mathcal{E} = -\frac{d\Phi_B}{dt} \). The magnetic flux (\(\Phi_B\)) through the ring is given by the product of the magnetic field strength (\(B\)) and the area (\(A\)) it passes through, assuming the field is uniform and perpendicular to the area: \(\Phi_B = B \cdot A\). In this specific problem, the magnetic field is given by \(B(t) = B_0 e^{-\alpha t}\) and the ring has a radius \(r\), so its area is \(A = \pi r^2\). Therefore, the magnetic flux is \(\Phi_B(t) = B_0 e^{-\alpha t} \cdot \pi r^2\). Applying Faraday’s Law, the induced EMF is \(\mathcal{E}(t) = -\frac{d}{dt}(B_0 \pi r^2 e^{-\alpha t})\). Differentiating the flux with respect to time, we get \(\frac{d\Phi_B}{dt} = B_0 \pi r^2 (-\alpha e^{-\alpha t})\). Thus, the induced EMF is \(\mathcal{E}(t) = – (-\alpha B_0 \pi r^2 e^{-\alpha t}) = \alpha B_0 \pi r^2 e^{-\alpha t}\). According to Ohm’s Law, the induced current (\(I\)) is related to the induced EMF by \(I = \frac{\mathcal{E}}{R}\), where \(R\) is the resistance of the ring. Therefore, the induced current is \(I(t) = \frac{\alpha B_0 \pi r^2 e^{-\alpha t}}{R}\). The question asks about the *magnitude* of the induced current. The magnitude of the induced current is therefore \(\frac{\alpha B_0 \pi r^2 e^{-\alpha t}}{R}\). This demonstrates how a changing magnetic field induces a current in a conductor, a core concept in electrical engineering and physics, directly relevant to understanding phenomena like eddy currents and transformer operation, which are foundational for many technologies studied at ESIGELEC IRSEEM Higher School of Engineering. The exponential decay of the magnetic field leads to an exponentially decaying induced current, highlighting the dynamic nature of electromagnetic interactions.

The core principle being tested here is the understanding of signal-to-noise ratio (SNR) in the context of digital communication systems, a fundamental concept in electrical engineering and telecommunications, areas of significant focus at ESIGELEC IRSEEM Higher School of Engineering. The signal power \(P_s\) is given as 10 milliwatts, which is \(10 \times 10^{-3}\) Watts. The noise power \(P_n\) is given as 0.5 milliwatts, which is \(0.5 \times 10^{-3}\) Watts. The signal-to-noise ratio (SNR) is defined as the ratio of signal power to noise power: \[ \text{SNR} = \frac{P_s}{P_n} \] Substituting the given values: \[ \text{SNR} = \frac{10 \times 10^{-3} \text{ W}}{0.5 \times 10^{-3} \text{ W}} \] \[ \text{SNR} = \frac{10}{0.5} \] \[ \text{SNR} = 20 \] To express this in decibels (dB), the formula is: \[ \text{SNR}_{\text{dB}} = 10 \log_{10} \left( \frac{P_s}{P_n} \right) \] \[ \text{SNR}_{\text{dB}} = 10 \log_{10}(20) \] Using a calculator, \( \log_{10}(20) \approx 1.301 \). \[ \text{SNR}_{\text{dB}} \approx 10 \times 1.301 \] \[ \text{SNR}_{\text{dB}} \approx 13.01 \text{ dB} \] This calculation demonstrates the direct relationship between signal power, noise power, and the resulting SNR, a critical metric for evaluating the quality and reliability of communication channels. A higher SNR indicates a stronger signal relative to the background noise, which is essential for accurate data transmission and reception, a key concern in fields like embedded systems and networked communications studied at ESIGELEC IRSEEM. Understanding how to quantify and improve SNR is vital for designing robust communication protocols and systems that can operate effectively in real-world environments characterized by interference and signal degradation. This question probes the foundational understanding of a concept that underpins many advanced topics in signal processing and telecommunications engineering.

The question probes the understanding of ethical considerations in the development and deployment of AI systems, a core tenet of responsible engineering education at ESIGELEC IRSEEM Higher School of Engineering. Specifically, it addresses the challenge of bias amplification in machine learning models trained on historical data. Consider a scenario where an AI system designed to assist in university admissions at ESIGELEC IRSEEM Higher School of Engineering is trained on historical applicant data. This data, unfortunately, reflects past societal biases, leading to a disproportionately lower acceptance rate for certain demographic groups. The AI, by learning patterns from this biased data, might inadvertently perpetuate or even amplify these existing inequalities. The fundamental ethical principle at play here is fairness and equity in algorithmic decision-making. While the AI might achieve high accuracy on the training data, its performance in real-world application would be compromised by its inherent bias. The goal of an AI engineer, particularly one educated at ESIGELEC IRSEEM Higher School of Engineering, is not just to build functional systems but to build *just* and *equitable* systems. Therefore, the most critical consideration when identifying and mitigating such bias is the *interpretability and explainability* of the AI’s decision-making process. Understanding *why* the AI makes certain recommendations allows engineers to pinpoint the sources of bias, whether they stem from data imbalances, feature selection, or algorithmic architecture. Without this transparency, attempts to correct bias would be akin to treating symptoms without understanding the disease. While other factors like data preprocessing, model retraining, and diverse development teams are crucial components of bias mitigation, they are all informed by and dependent on the ability to interpret the model’s behavior. If the model is a “black box,” it becomes exceedingly difficult to diagnose the root cause of the bias and implement effective corrective measures. Hence, the emphasis on interpretability is paramount for ensuring ethical AI development aligned with the rigorous standards of ESIGELEC IRSEEM Higher School of Engineering.

The core concept tested here is the understanding of the fundamental principles of electromagnetic induction and Faraday’s Law, particularly as applied to a scenario involving a changing magnetic flux through a coil. The question probes the ability to discern the primary driver of induced electromotive force (EMF) in a dynamic system. Faraday’s Law states that the induced EMF in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. Mathematically, this is expressed as \(\mathcal{E} = -\frac{d\Phi_B}{dt}\), where \(\mathcal{E}\) is the induced EMF and \(\Phi_B\) is the magnetic flux. Magnetic flux itself is defined as \(\Phi_B = \int \mathbf{B} \cdot d\mathbf{A}\), where \(\mathbf{B}\) is the magnetic field and \(d\mathbf{A}\) is an element of area. In the given scenario for ESIGELEC IRSEEM Higher School of Engineering Entrance Exam, a solenoid is connected to an AC voltage source, creating a time-varying magnetic field within it. This solenoid is then placed near a stationary secondary coil. The AC voltage source ensures that the current in the primary solenoid fluctuates sinusoidally, thereby generating a magnetic field that also fluctuates sinusoidally in time. This changing magnetic field permeates the secondary coil. According to Faraday’s Law, a changing magnetic flux through the secondary coil will induce an EMF in it. The magnitude of this induced EMF is directly proportional to the rate of change of the magnetic flux. Option a) correctly identifies that the induced EMF in the secondary coil is a direct consequence of the time-varying magnetic field produced by the primary solenoid. This time-varying field causes a change in magnetic flux through the secondary coil, which, by Faraday’s Law, induces an EMF. The AC voltage source is the ultimate origin of this phenomenon, driving the current that generates the fluctuating magnetic field. Option b) is incorrect because while the AC voltage source is essential for the operation, it is not the *direct* cause of the induced EMF in the secondary coil. The induced EMF arises from the *change* in magnetic flux, which is a consequence of the *changing magnetic field* generated by the primary coil’s current, which is driven by the voltage source. Option c) is incorrect. The number of turns in the primary solenoid influences the strength of the magnetic field it produces for a given current, and thus affects the magnitude of the flux change. However, the *fundamental principle* driving the induction is the change in flux itself, not the specific number of turns in the primary, unless that change in turns directly leads to a change in flux. The question asks for the primary cause of the induced EMF in the secondary coil. Option d) is incorrect. The presence of a secondary coil is necessary for the induced EMF to manifest as a measurable voltage or current in that coil. However, the induction phenomenon itself is driven by the changing magnetic flux, not merely the existence of the secondary coil. The secondary coil acts as the circuit through which the induced EMF is observed. Therefore, the most accurate and fundamental explanation for the induced EMF in the secondary coil is the time-varying magnetic field generated by the primary solenoid, which leads to a changing magnetic flux through the secondary coil.

The core concept here revolves around the principles of electromagnetic induction and Faraday’s Law, particularly as applied to a rotating coil in a non-uniform magnetic field, a scenario relevant to understanding AC generator principles taught at ESIGELEC IRSEEM Higher School of Engineering. The induced electromotive force (EMF) in a coil rotating in a magnetic field is proportional to the rate of change of magnetic flux through the coil. Faraday’s Law states that the induced EMF is given by \( \mathcal{E} = -N \frac{d\Phi_B}{dt} \), where \( N \) is the number of turns and \( \Phi_B \) is the magnetic flux. In this case, the magnetic field is not uniform and varies with position. If the coil rotates with angular velocity \( \omega \) in a magnetic field \( \mathbf{B} \), the flux through a single turn is \( \Phi_B = \int_S \mathbf{B} \cdot d\mathbf{A} \). For a coil of area \( A \) rotating in a uniform magnetic field \( B_0 \) perpendicular to the axis of rotation, the flux is \( \Phi_B = B_0 A \cos(\omega t) \), leading to an induced EMF \( \mathcal{E} = N B_0 A \omega \sin(\omega t) \). However, the problem specifies a non-uniform field where the magnetic field strength varies linearly with distance from the axis of rotation, say \( B(r) = kr \), and the coil is rotating in a plane perpendicular to this radial field. If the coil has an inner radius \( r_1 \) and outer radius \( r_2 \), and its width is \( w \), the area elements are \( dA = w \, dr \). The flux through the coil at any instant, assuming the field is radial and the coil rotates in a plane perpendicular to the field direction, would involve integrating the field over the area. However, the question implies a scenario where the *rate of change* of flux is the key. When a coil rotates, the flux changes. If the magnetic field itself is not uniform and its gradient is significant, the induced EMF will be affected by how the coil elements traverse regions of different field strengths and how the *effective* flux linkage changes. The question is designed to test the understanding that in a non-uniform field, the induced EMF is not simply proportional to the sine or cosine of the angle of rotation, but depends on the spatial distribution of the magnetic field and the geometry of the coil. Specifically, if the magnetic field strength varies linearly with radial distance from the axis of rotation, and the coil is rotating, the rate of change of flux will be influenced by this gradient. The induced EMF will be proportional to the product of the field gradient, the coil’s dimensions, and the angular velocity. The correct answer reflects this dependency on the field gradient and the coil’s radial extent. The induced EMF in such a scenario, considering a coil with radial width \( w \) and rotating with angular velocity \( \omega \), where the magnetic field is \( B(r) = kr \) and directed radially, would be \( \mathcal{E} = -N \frac{d}{dt} \int_{r_1}^{r_2} (kr) w \, dr \). Since \( k \) and \( w \) are constants and the integration is over \( r \), the flux is \( \Phi_B = Nkw \int_{r_1}^{r_2} r \, dr = Nkw \left[ \frac{r^2}{2} \right]_{r_1}^{r_2} = \frac{1}{2} Nkw (r_2^2 – r_1^2) \). This flux is constant with time if the coil is just rotating in a fixed radial field. However, the question implies a scenario where the *change* in flux is what matters, typically due to rotation. A more appropriate interpretation for a rotating coil in a radial field where the field strength varies with radial distance \( r \) as \( B(r) = kr \) and the coil is rotating in a plane perpendicular to the field direction would involve considering the flux linkage. If the coil is rotating, the flux through it changes. For a coil rotating in a plane, the flux is \( \Phi_B = \int \mathbf{B} \cdot d\mathbf{A} \). If \( \mathbf{B} \) is radial and the coil is rotating, the orientation of \( d\mathbf{A} \) relative to \( \mathbf{B} \) changes. However, if the coil is rotating in a plane and the field is radial, the flux linkage might not change in a simple sinusoidal way. Let’s re-evaluate the premise for a rotating coil in a radial field. If the coil is rotating about an axis, and the field is radial, the flux linkage will depend on the orientation. A more common scenario for induced EMF in non-uniform fields involves motion through the field. Consider a single conducting rod of length \( L \) moving with velocity \( v \) perpendicular to a magnetic field \( B \). The induced EMF is \( \mathcal{E} = BLv \). If the field varies with position, \( B(x) \), and the rod moves from \( x_1 \) to \( x_2 \), \( \mathcal{E} = \int_{x_1}^{x_2} B(x) v \, dx \). For a rotating coil in a radial field \( B(r) = kr \), where the coil has radial extent \( r_1 \) to \( r_2 \) and is rotating with angular velocity \( \omega \), the velocity of a segment at radius \( r \) is \( v = \omega r \). The induced EMF in a radial segment of length \( dr \) would be \( d\mathcal{E} = B(r) v \, dr = (kr)(\omega r) \, dr = k\omega r^2 \, dr \). Integrating this over the radial extent of the coil gives the total induced EMF: \( \mathcal{E} = \int_{r_1}^{r_2} k\omega r^2 \, dr = k\omega \left[ \frac{r^3}{3} \right]_{r_1}^{r_2} = \frac{1}{3} k\omega (r_2^3 – r_1^3) \). This result shows the induced EMF is proportional to the field gradient \( k \), the angular velocity \( \omega \), and a term dependent on the inner and outer radii of the coil. Therefore, the induced EMF is directly proportional to the field gradient \( k \). The question probes the fundamental understanding of electromagnetic induction in non-uniform fields, a concept crucial for advanced studies in electrical engineering and physics, particularly in areas like sensor design and electric machine principles, which are integral to the curriculum at ESIGELEC IRSEEM Higher School of Engineering. Faraday’s Law of Induction is the bedrock principle, stating that a changing magnetic flux through a circuit induces an electromotive force (EMF). When the magnetic field is not uniform, the calculation of magnetic flux becomes more complex, requiring integration over the area of the coil. Furthermore, if the coil is in motion, the rate of change of flux is influenced by both the temporal variation of the field (if any) and the spatial variation of the field as the coil moves through it. In this specific scenario, the magnetic field strength increases linearly with radial distance from a central axis, \( B(r) = kr \), and the coil is rotating in a plane perpendicular to this radial field. For a rotating coil, the velocity of different parts of the coil varies with their radial distance from the axis of rotation. A segment of the coil at radius \( r \) will have a tangential velocity \( v = \omega r \), where \( \omega \) is the angular velocity. The induced EMF in a small radial segment of the coil of length \( dr \) can be thought of as \( d\mathcal{E} = B(r) v \, dr \), assuming the segment is perpendicular to both the field and its velocity. Substituting the given relationships, \( d\mathcal{E} = (kr)(\omega r) \, dr = k\omega r^2 \, dr \). To find the total induced EMF across the coil, we integrate this expression over the radial extent of the coil, from its inner radius \( r_1 \) to its outer radius \( r_2 \): \( \mathcal{E} = \int_{r_1}^{r_2} k\omega r^2 \, dr \). Performing the integration, we get \( \mathcal{E} = k\omega \left[ \frac{r^3}{3} \right]_{r_1}^{r_2} = \frac{1}{3} k\omega (r_2^3 – r_1^3) \). This result clearly demonstrates that the induced EMF is directly proportional to the magnetic field gradient \( k \). This understanding is vital for students at ESIGELEC IRSEEM Higher School of Engineering as it relates to the design and analysis of rotating electrical machinery, where non-uniform magnetic fields can significantly impact performance and efficiency. It also highlights the importance of spatial field distribution in electromagnetic phenomena, a key area of study in advanced electromagnetics.

The scenario describes a system where a signal is processed through a series of filters. The first filter is a low-pass filter with a cutoff frequency \(f_c = 1000 \, \text{Hz}\). The second filter is a high-pass filter with a cutoff frequency \(f_o = 500 \, \text{Hz}\). A signal with a fundamental frequency of \(f_{fund} = 750 \, \text{Hz}\) is applied. A low-pass filter attenuates frequencies above its cutoff frequency. Therefore, the low-pass filter with \(f_c = 1000 \, \text{Hz}\) will allow frequencies up to \(1000 \, \text{Hz}\) to pass through with minimal attenuation. The signal’s fundamental frequency of \(750 \, \text{Hz}\) is below this cutoff, so it will pass through the first filter. A high-pass filter attenuates frequencies below its cutoff frequency. The high-pass filter with \(f_o = 500 \, \text{Hz}\) will attenuate frequencies below \(500 \, \text{Hz}\). The signal’s fundamental frequency of \(750 \, \text{Hz}\) is above this cutoff, so it will also pass through the second filter. When a signal passes through a cascade of filters, the overall effect is the combined filtering characteristics. In this case, the signal at \(750 \, \text{Hz}\) is above the high-pass cutoff (\(750 \, \text{Hz} > 500 \, \text{Hz}\)) and below the low-pass cutoff (\(750 \, \text{Hz} < 1000 \, \text{Hz}\)). This means the signal falls within the passband of both filters. The combination of a low-pass filter followed by a high-pass filter, where the high-pass cutoff is lower than the low-pass cutoff, creates a band-pass filtering effect. The signal's fundamental frequency lies within this effective passband. Therefore, the signal will be transmitted through the entire system with minimal attenuation, preserving its fundamental frequency. The question asks about the fundamental frequency of the signal *after* passing through both filters. Since \(750 \, \text{Hz}\) is within the passband defined by the two filters, the fundamental frequency remains \(750 \, \text{Hz}\).

The core principle tested here is the understanding of the Nyquist-Shannon sampling theorem and its implications for signal reconstruction. The theorem states that to perfectly reconstruct a signal from its samples, the sampling frequency (\(f_s\)) must be at least twice the highest frequency component (\(f_{max}\)) present in the signal. This minimum sampling rate is known as the Nyquist rate, \(f_{Nyquist} = 2f_{max}\). In this scenario, the analog signal contains frequency components up to \(15 \text{ kHz}\). Therefore, \(f_{max} = 15 \text{ kHz}\). According to the Nyquist-Shannon sampling theorem, the minimum sampling frequency required for perfect reconstruction is \(f_s \ge 2 \times f_{max}\). Calculation: Minimum \(f_s = 2 \times 15 \text{ kHz} = 30 \text{ kHz}\). The question asks for the *minimum* sampling frequency required. Any sampling frequency below \(30 \text{ kHz}\) would lead to aliasing, where higher frequencies masquerade as lower frequencies, making accurate reconstruction impossible. Sampling at exactly \(30 \text{ kHz}\) satisfies the theorem for perfect reconstruction. Sampling at a higher frequency, such as \(40 \text{ kHz}\), would also allow for reconstruction, but the question specifically asks for the minimum required. Therefore, \(30 \text{ kHz}\) is the correct answer. This concept is fundamental in digital signal processing, a key area within the curriculum at ESIGELEC IRSEEM Higher School of Engineering, particularly for students specializing in areas like telecommunications, embedded systems, and signal processing. Understanding aliasing and the conditions for perfect reconstruction is crucial for designing effective digital systems that interface with the analog world, ensuring data integrity and signal fidelity.

The core principle tested here is the understanding of signal quantization and its impact on signal-to-quantization-noise ratio (SQNR). For a uniform quantizer, the SQNR is directly proportional to the number of quantization levels raised to a power related to the number of bits per sample. Specifically, for a sinusoidal input signal, the SQNR in decibels (dB) is given by the formula: \( \text{SQNR}_{\text{dB}} = 6.02n + 1.76 \), where \( n \) is the number of bits per sample. In this scenario, the ESIGELEC IRSEEM Higher School of Engineering Entrance Exam is evaluating the candidate’s ability to reason about the trade-offs in digital signal processing. When the number of bits per sample is reduced from 12 bits to 10 bits, the number of quantization levels is halved for each bit reduction. This means the dynamic range is divided by \( 2^{12-10} = 2^2 = 4 \). Let \( n_1 = 12 \) bits and \( n_2 = 10 \) bits. The SQNR for \( n_1 \) bits is \( \text{SQNR}_1 = 6.02(12) + 1.76 = 72.24 + 1.76 = 74 \text{ dB} \). The SQNR for \( n_2 \) bits is \( \text{SQNR}_2 = 6.02(10) + 1.76 = 60.2 + 1.76 = 61.96 \text{ dB} \). The reduction in SQNR is \( \Delta \text{SQNR} = \text{SQNR}_1 – \text{SQNR}_2 = 74 \text{ dB} – 61.96 \text{ dB} = 12.04 \text{ dB} \). Alternatively, each bit reduction reduces the SQNR by approximately 6 dB. A reduction from 12 bits to 10 bits is a reduction of 2 bits. Therefore, the expected reduction in SQNR is approximately \( 2 \times 6.02 \text{ dB} \approx 12.04 \text{ dB} \). This decrease in SQNR directly impacts the fidelity of the digitized signal, making it more susceptible to quantization errors, which is a fundamental concept in digital communications and signal processing taught at institutions like ESIGELEC IRSEEM Higher School of Engineering Entrance Exam. Understanding this relationship is crucial for designing efficient and accurate digital systems, a key focus in the engineering curriculum.

The core principle being tested here is the understanding of signal-to-noise ratio (SNR) in the context of digital communication systems, a fundamental concept within ESIGELEC IRSEEM’s curriculum, particularly in areas like telecommunications and signal processing. SNR is defined as the ratio of the power of a signal to the power of background noise. Mathematically, it’s often expressed in decibels (dB) as \(SNR_{dB} = 10 \log_{10} \left(\frac{P_{signal}}{P_{noise}}\right)\). Consider a scenario where a communication system is operating with a certain signal power \(P_s\) and noise power \(P_n\). The initial SNR is \(SNR_1 = \frac{P_s}{P_n}\). If the signal power is increased by a factor of 4, the new signal power becomes \(4P_s\). The noise power remains unchanged at \(P_n\). The new SNR, \(SNR_2\), is therefore \(SNR_2 = \frac{4P_s}{P_n} = 4 \times \frac{P_s}{P_n} = 4 \times SNR_1\). To express this change in decibels, we look at the difference in SNR: \(SNR_{dB,2} – SNR_{dB,1} = 10 \log_{10} \left(\frac{4P_s}{P_n}\right) – 10 \log_{10} \left(\frac{P_s}{P_n}\right)\) \( = 10 \left[ \log_{10} \left(\frac{4P_s}{P_n}\right) – \log_{10} \left(\frac{P_s}{P_n}\right) \right]\) \( = 10 \log_{10} \left( \frac{4P_s/P_n}{P_s/P_n} \right)\) \( = 10 \log_{10} (4)\) Calculating \(10 \log_{10} (4)\): \( \log_{10} (4) \approx 0.60206 \) \( 10 \times 0.60206 \approx 6.0206 \) Therefore, an increase in signal power by a factor of 4 results in an increase in SNR of approximately 6.02 dB. This concept is crucial for understanding link budgets, error rates, and the overall performance of wireless and wired communication systems, directly relevant to ESIGELEC IRSEEM’s advanced studies in these fields. A higher SNR generally leads to more reliable data transmission and fewer errors, which is a primary objective in designing efficient communication protocols. Understanding how power variations affect SNR is fundamental to optimizing system performance and meeting stringent quality-of-service requirements.

The question probes the understanding of the fundamental principles of electromagnetic induction and Faraday’s Law, specifically as applied to a scenario involving a changing magnetic flux within a conductor. The core concept is that a changing magnetic flux through a closed circuit induces an electromotive force (EMF), which in turn drives a current. Faraday’s Law quantifies this relationship: \( \mathcal{E} = -\frac{d\Phi_B}{dt} \), where \( \mathcal{E} \) is the induced EMF and \( \Phi_B \) is the magnetic flux. In this context, the magnetic flux is given by \( \Phi_B = \int \mathbf{B} \cdot d\mathbf{A} \). The problem describes a solenoid with a uniform magnetic field \( \mathbf{B} \) directed along its axis, and the magnetic field strength is changing linearly with time. The rate of change of the magnetic field is \( \frac{dB}{dt} = 0.5 \, \text{T/s} \). For a solenoid of cross-sectional area \( A \), the magnetic flux through each turn is \( \Phi_B = B \cdot A \). Therefore, the rate of change of magnetic flux through each turn is \( \frac{d\Phi_B}{dt} = A \frac{dB}{dt} \). The induced EMF in a single turn is \( \mathcal{E}_{\text{turn}} = -A \frac{dB}{dt} \). The total induced EMF in the coil with \( N \) turns is \( \mathcal{E}_{\text{total}} = N \mathcal{E}_{\text{turn}} = -N A \frac{dB}{dt} \). The question specifies a solenoid with 100 turns, a radius of 5 cm (which means a cross-sectional area \( A = \pi r^2 = \pi (0.05 \, \text{m})^2 = 0.0025\pi \, \text{m}^2 \)), and a rate of change of magnetic field of 0.5 T/s. Substituting these values, the magnitude of the induced EMF is \( |\mathcal{E}_{\text{total}}| = 100 \times (0.0025\pi \, \text{m}^2) \times (0.5 \, \text{T/s}) = 0.125\pi \, \text{V} \). This induced EMF drives a current in the coil. The question asks about the nature of the induced current and its relationship to the changing magnetic field. Lenz’s Law, which is incorporated in the negative sign of Faraday’s Law, states that the direction of the induced current will be such that it opposes the change in magnetic flux that produced it. Since the magnetic field is increasing along the axis, the induced current will create its own magnetic field in the opposite direction to oppose this increase. This implies that if the external field is directed, for example, upwards, the induced field will be downwards, and by the right-hand rule, the current will flow in a specific direction around the solenoid. The question is designed to test the understanding of how a changing magnetic field induces an EMF and subsequently a current, and how the properties of the coil (number of turns, area) and the rate of change of the field influence the magnitude of this induced EMF. The core principle is the conversion of magnetic energy into electrical energy due to a dynamic change in the magnetic environment, a fundamental concept in electromagnetism relevant to many engineering applications studied at ESIGELEC IRSEEM Higher School of Engineering. The magnitude of the induced EMF is directly proportional to the number of turns and the rate of change of the magnetic flux, and the area through which the flux passes.

The core concept here is understanding the interplay between signal bandwidth, data rate, and channel capacity in digital communications, a fundamental area within ESIGELEC IRSEEM’s curriculum, particularly in signal processing and telecommunications. Shannon’s capacity theorem provides the theoretical upper bound for reliable data transmission over a noisy channel. The formula for channel capacity \(C\) is given by \(C = B \log_2(1 + S/N)\), where \(B\) is the bandwidth in Hertz and \(S/N\) is the signal-to-noise ratio (SNR). In this scenario, we are given a channel with a bandwidth \(B = 4 \text{ kHz}\) and an SNR of \(1000\). We need to determine the maximum achievable data rate (channel capacity) in bits per second (bps). Calculation: \(C = B \log_2(1 + S/N)\) \(C = 4000 \text{ Hz} \times \log_2(1 + 1000)\) \(C = 4000 \times \log_2(1001)\) To calculate \(\log_2(1001)\), we can use the change of base formula: \(\log_2(x) = \frac{\log_{10}(x)}{\log_{10}(2)}\) or \(\log_2(x) = \frac{\ln(x)}{\ln(2)}\). \(\log_2(1001) \approx \frac{\log_{10}(1001)}{\log_{10}(2)} \approx \frac{3.0004}{0.3010} \approx 9.968\) Alternatively, recognizing that \(2^{10} = 1024\), \(\log_2(1001)\) will be slightly less than 10. A more precise calculation yields approximately 9.968. \(C \approx 4000 \times 9.968\) \(C \approx 39872 \text{ bps}\) This value represents the theoretical maximum data rate. In practice, achieving this capacity requires sophisticated coding and modulation schemes, which are key areas of study at ESIGELEC IRSEEM. The question probes the understanding of this fundamental limit and its implications for designing efficient communication systems. The ability to apply Shannon’s theorem is crucial for students aiming to work in fields like wireless communication, network engineering, and digital signal processing, all of which are central to the engineering programs offered at ESIGELEC IRSEEM. Understanding this limit helps in setting realistic expectations for system performance and in evaluating the effectiveness of different communication technologies.

The core concept tested here is the understanding of signal-to-noise ratio (SNR) and its impact on data transmission reliability, a fundamental principle in telecommunications and signal processing, areas of significant focus at ESIGELEC IRSEEM Higher School of Engineering. While no direct calculation is required, the question probes the qualitative understanding of how signal degradation affects the ability to discern meaningful information. A higher SNR indicates a stronger signal relative to background noise, leading to fewer errors in data reception. Conversely, a lower SNR means the noise is more prominent, making it harder to accurately decode the transmitted information. Therefore, to improve the reliability of data transmission in a noisy environment, the primary objective is to increase the SNR. This can be achieved through various engineering techniques such as signal amplification, noise reduction filtering, or employing more robust modulation schemes. The question implicitly asks about the most direct consequence of a degraded signal in terms of its interpretability.

The core principle being tested here is the understanding of signal-to-noise ratio (SNR) in the context of digital signal processing and its impact on data integrity, a fundamental concept within ESIGELEC IRSEEM’s electrical engineering and computer science programs. While no explicit calculation is required for the final answer selection, the underlying concept involves understanding how noise degrades a signal. A higher SNR indicates a stronger signal relative to background noise, leading to more reliable data transmission and processing. Conversely, a lower SNR implies that noise is a significant factor, potentially corrupting the signal and leading to errors in interpretation or reconstruction. The question probes the candidate’s ability to connect this theoretical concept to practical implications in digital systems, such as the fidelity of sensor readings or the accuracy of communication protocols. The ability to discern the most critical factor for maintaining data integrity in the face of inherent system imperfections is key. A robust understanding of how noise affects signal representation and the strategies employed to mitigate its impact, such as error correction codes or advanced filtering techniques, is crucial for success in advanced engineering studies at ESIGELEC IRSEEM.

The core concept tested here is the understanding of signal-to-noise ratio (SNR) and its impact on data transmission reliability, a fundamental aspect in fields like telecommunications and embedded systems, both integral to ESIGELEC IRSEEM’s curriculum. A higher SNR indicates a stronger signal relative to background noise, leading to fewer errors in data reception. Conversely, a lower SNR degrades the signal quality, increasing the probability of bit errors. Consider a digital communication system transmitting data at a rate \(R\) bits per second. The channel is characterized by a bandwidth \(B\) Hertz and an average noise power \(N\). The transmitted signal power is \(S\). The Shannon-Hartley theorem provides the theoretical maximum data rate \(C\) for a given channel: \(C = B \log_2(1 + \frac{S}{N})\). The term \(\frac{S}{N}\) represents the SNR. If the SNR is \(10\), the maximum achievable data rate is \(C_1 = B \log_2(1 + 10) = B \log_2(11)\). If the SNR is increased to \(20\), the maximum achievable data rate becomes \(C_2 = B \log_2(1 + 20) = B \log_2(21)\). The question asks about the *relative* improvement in data transmission reliability. While the Shannon-Hartley theorem quantifies the *theoretical maximum rate*, practical reliability is directly correlated with the SNR. An increase in SNR from \(10\) to \(20\) means the signal power is now \(20/10 = 2\) times stronger relative to the noise. This doubling of the SNR, in a logarithmic scale, leads to a significant improvement in the probability of error. Specifically, the term \(\log_2(1 + \text{SNR})\) in the Shannon-Hartley theorem shows that as SNR increases, the capacity increases, but the relationship is logarithmic. However, the question is about *reliability*, which is more directly tied to the signal’s dominance over noise. A higher SNR directly translates to a lower probability of bit errors. Therefore, doubling the SNR from \(10\) to \(20\) implies a substantial enhancement in the signal’s ability to be distinguished from noise, leading to a more robust and reliable transmission. This improvement is not a linear doubling of data rate but a significant reduction in error probability, making the transmission more dependable. The most accurate description of this improvement, without resorting to specific error rate calculations which depend on modulation schemes, is a substantial increase in the signal’s robustness against interference.

The core principle being tested here is the understanding of **electromagnetic induction** and its application in energy transfer, specifically in the context of wireless power transfer systems, a field relevant to ESIGELEC IRSEEM’s focus on embedded systems and connected objects. The scenario describes a primary coil (transmitter) and a secondary coil (receiver) separated by a non-conductive medium. When an alternating current flows through the primary coil, it generates a time-varying magnetic field. This changing magnetic field, according to Faraday’s Law of Induction, induces an electromotive force (EMF) in any conductor that it links. In this case, the secondary coil acts as that conductor. The induced EMF in the secondary coil drives a current, thereby transferring energy wirelessly. The efficiency of this energy transfer is influenced by several factors, including the frequency of the alternating current, the geometry and proximity of the coils, and the magnetic coupling between them. A higher frequency generally leads to a stronger induced EMF for a given rate of change of magnetic flux, but also introduces losses due to eddy currents and skin effect. The magnetic coupling, quantified by the coupling coefficient \(k\), is crucial; it represents the fraction of the magnetic flux produced by the primary coil that actually links with the secondary coil. A higher coupling coefficient signifies more efficient energy transfer. The question asks about the fundamental mechanism enabling this transfer. The induced voltage in the secondary coil is directly proportional to the rate of change of magnetic flux linking it, which is a direct consequence of the time-varying magnetic field generated by the primary coil. Therefore, the presence of a time-varying magnetic field is the indispensable prerequisite for inducing a voltage and consequently transferring power.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning aliasing and sampling. When a continuous-time signal \(x(t)\) is sampled at a rate \(f_s\), the resulting discrete-time signal \(x[n] = x(nT)\) where \(T = 1/f_s\), can exhibit aliasing if the signal contains frequency components above \(f_s/2\). The Nyquist-Shannon sampling theorem states that to perfectly reconstruct a band-limited signal, the sampling frequency must be at least twice the highest frequency component in the signal. If this condition is not met, higher frequencies in the original signal “fold back” into the lower frequency range, distorting the sampled signal. Consider a signal with a maximum frequency component of \(f_{max}\). If this signal is sampled at \(f_s\), and \(f_{max} > f_s/2\), aliasing will occur. The aliased frequency \(f_{alias}\) for a frequency \(f > f_s/2\) is given by \(f_{alias} = |f – k f_s|\) for some integer \(k\) such that \(f_{alias} \le f_s/2\). The smallest positive value of \(f_{alias}\) occurs when \(k\) is chosen such that \(f – k f_s\) is closest to zero but still within the \([0, f_s/2]\) range. For a frequency \(f\), the aliased frequency is \(f \pmod{f_s}\), and if this result is greater than \(f_s/2\), the aliased frequency is \(f_s – (f \pmod{f_s})\). In the context of the ESIGELEC IRSEEM Higher School of Engineering’s curriculum, understanding sampling and aliasing is crucial for various fields including telecommunications, control systems, and embedded systems, where signals are frequently digitized. The ability to identify and mitigate aliasing through appropriate sampling rates or anti-aliasing filters is a core competency. The question tests the candidate’s ability to apply the Nyquist criterion and understand the consequences of violating it, which is a foundational concept in signal processing education at institutions like ESIGELEC IRSEEM. This knowledge is essential for designing robust and accurate digital systems.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its practical implications in analog-to-digital conversion (ADC). The scenario describes a system designed to capture audio signals for analysis at ESIGELEC IRSEEM Higher School of Engineering. The audio signal has a maximum frequency component of 15 kHz. The Nyquist-Shannon sampling theorem states that to perfectly reconstruct an analog signal from its sampled version, the sampling frequency (\(f_s\)) must be at least twice the highest frequency component (\(f_{max}\)) present in the signal. This minimum sampling rate is known as the Nyquist rate, given by \(f_{Nyquist} = 2 \times f_{max}\). In this case, \(f_{max} = 15 \text{ kHz}\). Therefore, the minimum sampling frequency required to avoid aliasing and ensure accurate reconstruction is: \(f_s \ge 2 \times 15 \text{ kHz}\) \(f_s \ge 30 \text{ kHz}\) The question asks about the consequence of sampling at a rate *below* this minimum requirement. Sampling below the Nyquist rate leads to a phenomenon called aliasing. Aliasing occurs when high-frequency components in the analog signal are incorrectly represented as lower frequencies in the sampled digital signal. This distortion is irreversible and corrupts the integrity of the digital representation, making accurate reconstruction impossible. The lower frequencies masquerade as higher frequencies, and vice-versa, leading to a misinterpretation of the original signal’s spectral content. This is a critical concept for students at ESIGELEC IRSEEM Higher School of Engineering, as it directly impacts the design and performance of communication systems, sensor data acquisition, and audio processing applications, all of which are integral to various engineering disciplines taught at the institution. Understanding and preventing aliasing is paramount for reliable signal processing.

The core principle tested here relates to the fundamental trade-offs in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its practical implications in real-world systems like those developed at ESIGELEC IRSEEM Higher School of Engineering. The theorem states that to perfectly reconstruct a signal, the sampling frequency (\(f_s\)) must be at least twice the highest frequency component (\(f_{max}\)) present in the signal, i.e., \(f_s \ge 2f_{max}\). This minimum sampling rate is known as the Nyquist rate. Aliasing occurs when a signal is sampled at a rate lower than the Nyquist rate. In such cases, higher frequencies in the analog signal are incorrectly interpreted as lower frequencies in the sampled digital signal. This phenomenon is irreversible; once aliasing has occurred, the original high-frequency information is lost and cannot be recovered. To prevent aliasing, an anti-aliasing filter (typically a low-pass filter) is applied to the analog signal *before* sampling. This filter attenuates or removes frequencies above a certain cutoff frequency, ensuring that only frequencies below \(f_s/2\) are present in the signal when it is sampled. If a signal containing frequencies up to \(15\) kHz is to be sampled without aliasing, the minimum sampling frequency required is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). If the sampling frequency is set to \(25 \text{ kHz}\), frequencies above \(25 \text{ kHz} / 2 = 12.5 \text{ kHz}\) will be subject to aliasing. Specifically, the \(15 \text{ kHz}\) component would be aliased to a frequency of \(|15 \text{ kHz} – 25 \text{ kHz}| = 10 \text{ kHz}\). Therefore, to accurately represent the \(15 \text{ kHz}\) component, the sampling frequency must be at least \(30 \text{ kHz}\). Using a sampling frequency of \(25 \text{ kHz}\) is insufficient. The most effective strategy to ensure accurate digital representation of signals with components up to \(15 \text{ kHz}\) when a \(25 \text{ kHz}\) sampling rate is mandated (perhaps due to hardware constraints or processing limitations) is to pre-filter the signal. This pre-filtering, using an anti-aliasing filter with a cutoff frequency below \(12.5 \text{ kHz}\) (and ideally close to \(12.5 \text{ kHz}\) but below it, to allow for practical filter roll-off), would remove the \(15 \text{ kHz}\) component before sampling, preventing it from causing aliasing. This ensures that the sampled signal accurately reflects the components that are within the valid bandwidth of the sampling rate.

The core principle being tested here is the understanding of signal-to-noise ratio (SNR) in the context of digital communication systems, a fundamental concept within ESIGELEC IRSEEM’s curriculum, particularly in areas like telecommunications and signal processing. The question probes the relationship between the power of the desired signal, the power of the interfering noise, and the overall data transmission rate. The signal power is given as \(P_s = 10^{-12}\) Watts. The noise power spectral density is given as \(N_0 = 10^{-15}\) Watts/Hz. The bandwidth of the communication channel is \(B = 10^5\) Hz. First, we calculate the total noise power within the given bandwidth: Total Noise Power \(P_n = N_0 \times B\) \(P_n = (10^{-15} \text{ W/Hz}) \times (10^5 \text{ Hz})\) \(P_n = 10^{-10}\) Watts Next, we calculate the Signal-to-Noise Ratio (SNR) in linear terms: SNR (linear) = \(P_s / P_n\) SNR (linear) = \(10^{-12} \text{ W} / 10^{-10} \text{ W}\) SNR (linear) = \(10^{-2}\) To express SNR in decibels (dB), we use the formula: SNR (dB) = \(10 \times \log_{10}(\text{SNR (linear)})\) SNR (dB) = \(10 \times \log_{10}(10^{-2})\) SNR (dB) = \(10 \times (-2)\) SNR (dB) = \(-20\) dB The Shannon-Hartley theorem states that the maximum achievable data rate \(C\) (channel capacity) is given by: \(C = B \log_2(1 + \text{SNR})\) However, the question asks about the *impact* of a reduced SNR on the *potential* for reliable communication, not the absolute maximum rate. A negative SNR in dB indicates that the noise power is significantly greater than the signal power. In practical terms, a negative SNR means the signal is deeply buried in noise, making reliable detection and decoding extremely challenging, if not impossible, without advanced error correction coding. The fundamental limit for reliable communication, as per Shannon’s theorem, is approached when the SNR is positive. A highly negative SNR implies that the channel is severely degraded, and achieving any meaningful data rate would require a substantial increase in signal power or a drastic reduction in bandwidth, or the use of extremely robust but low-rate coding schemes. The question is designed to assess the understanding that a very low (negative in dB) SNR fundamentally limits the feasibility of high-speed, reliable data transmission, a concept central to the information theory studied at ESIGELEC IRSEEM. The ability to interpret this negative dB value as a severe degradation of the communication channel is key.

The core principle being tested here is the understanding of electromagnetic wave propagation in a dielectric medium and its relation to the material properties. For a plane electromagnetic wave propagating in a homogeneous, isotropic, linear dielectric medium, the wave vector \( \vec{k} \) is related to the angular frequency \( \omega \), the permittivity \( \epsilon \), and the permeability \( \mu \) of the medium by the dispersion relation: \( k = \omega \sqrt{\mu \epsilon} \). The phase velocity \( v_p \) of the wave in the medium is given by \( v_p = \frac{\omega}{k} \). Substituting the expression for \( k \), we get \( v_p = \frac{\omega}{\omega \sqrt{\mu \epsilon}} = \frac{1}{\sqrt{\mu \epsilon}} \). In a non-magnetic dielectric material, the relative permeability \( \mu_r \) is approximately 1, so \( \mu \approx \mu_0 \), where \( \mu_0 \) is the permeability of free space. The permittivity of the medium is \( \epsilon = \epsilon_r \epsilon_0 \), where \( \epsilon_r \) is the relative permittivity (dielectric constant) and \( \epsilon_0 \) is the permittivity of free space. Therefore, the phase velocity becomes \( v_p = \frac{1}{\sqrt{\mu_0 \epsilon_r \epsilon_0}} = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \frac{1}{\sqrt{\epsilon_r}} \). We know that the speed of light in vacuum is \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \). Thus, \( v_p = \frac{c}{\sqrt{\epsilon_r}} \). The question describes a scenario where an electromagnetic wave, initially propagating in free space with speed \( c \), enters a dielectric material with a relative permittivity \( \epsilon_r = 4 \). The speed of the wave in the dielectric medium will be \( v_p = \frac{c}{\sqrt{4}} = \frac{c}{2} \). This reduction in speed is a fundamental consequence of the interaction of the electromagnetic field with the polarized molecules of the dielectric material, which effectively slows down the wave’s propagation. This concept is crucial in understanding phenomena like refraction and the design of optical and microwave components, areas of significant interest within ESIGELEC IRSEEM Higher School of Engineering’s curriculum, particularly in fields like telecommunications and embedded systems. The ability to predict and analyze wave behavior in different media is a foundational skill for engineers working with wave phenomena.

The core principle tested here is the understanding of signal-to-noise ratio (SNR) in the context of digital communication systems, a fundamental concept within ESIGELEC IRSEEM’s curriculum, particularly in areas like telecommunications and signal processing. The question probes the candidate’s ability to discern how different modulation schemes, when subjected to equivalent channel conditions, impact the effective data rate achievable for a given acceptable error rate. Consider a scenario where two digital modulation schemes, Phase-Shift Keying (PSK) and Quadrature Amplitude Modulation (QAM), are evaluated for transmission over a noisy channel. Both schemes are designed to achieve a Bit Error Rate (BER) of \(10^{-5}\). The channel is characterized by a fixed bandwidth \(B\) and a fixed noise power spectral density \(N_0\). The transmitted signal power is also kept constant for both schemes. According to Shannon’s channel capacity theorem, the maximum achievable data rate \(C\) over a channel with bandwidth \(B\) and SNR \(\gamma\) is given by \(C = B \log_2(1 + \gamma)\). While this theorem provides an upper bound, practical modulation schemes aim to approach this capacity. For a given BER target, different modulation schemes require different minimum SNRs. For instance, to achieve a BER of \(10^{-5}\), M-PSK typically requires a higher SNR per bit than M-QAM for the same number of bits per symbol, especially as M increases. However, the question implies a comparison of *equivalent* performance in terms of BER and signal power, but with different spectral efficiencies. If we assume that for the target BER, a specific modulation scheme (e.g., a higher-order QAM) can achieve a higher spectral efficiency (bits per second per Hertz) than another scheme (e.g., a lower-order PSK) under the same power and noise conditions, it means that the QAM scheme can pack more information into the same bandwidth. This is because QAM, by varying both amplitude and phase, can represent more distinct symbols than PSK (which only varies phase) for a given constellation size, or can achieve the same number of bits per symbol with a lower SNR requirement compared to PSK. Therefore, if both are operating at the same power and noise level, and achieving the same BER, the scheme that can represent more bits per symbol or achieve the target BER with less power per bit will yield a higher data rate. In this context, QAM generally offers better spectral efficiency than PSK for a given BER, allowing for a higher data rate within the same bandwidth and power constraints. The question is designed to test the understanding that different modulation techniques have different trade-offs between spectral efficiency, power efficiency, and robustness to noise, and that for equivalent BER and power, a more spectrally efficient scheme will yield a higher data rate.

The core principle being tested here is the understanding of how the fundamental theorem of calculus relates to the interpretation of rates of change in a physical system, specifically in the context of electromagnetism as studied at ESIGELEC IRSEEM Higher School of Engineering. Faraday’s Law of Induction states that the induced electromotive force (EMF) in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. Mathematically, this is expressed as \(\mathcal{E} = -\frac{d\Phi_B}{dt}\). Magnetic flux (\(\Phi_B\)) is defined as the integral of the magnetic field (\(\mathbf{B}\)) over a surface (\(S\)), \(\Phi_B = \int_S \mathbf{B} \cdot d\mathbf{A}\). Therefore, the induced EMF is directly proportional to how quickly the magnetic field passing through the loop is changing. If the magnetic field is constant over time, its derivative with respect to time is zero, meaning no EMF is induced. Conversely, a rapidly changing magnetic field will induce a larger EMF. The question asks about a scenario where a conductor loop is placed in a magnetic field that is *increasing uniformly*. This means the rate of change of the magnetic field, and consequently the magnetic flux, is constant and positive. According to Faraday’s Law, this constant positive rate of change of flux will induce a constant, non-zero EMF. The direction of this induced EMF (and hence the induced current) is given by Lenz’s Law, which states it opposes the change in magnetic flux. However, the question specifically asks about the *magnitude* of the induced EMF, which is directly proportional to the rate of change of flux. Since the magnetic field is increasing uniformly, the rate of change of flux is constant, leading to a constant induced EMF. This concept is foundational for understanding electromagnetic induction, a key area in electrical engineering and applied physics relevant to ESIGELEC IRSEEM’s curriculum, particularly in courses dealing with electrical machines, sensors, and signal processing. The ability to interpret the implications of a changing magnetic field on induced voltage is crucial for designing and analyzing electromagnetic devices.