ESIEE Paris Entrance Exam Set

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its implications for signal reconstruction. 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 strictly greater than twice the maximum frequency component of the signal. This minimum sampling rate is known as the Nyquist rate, which is \(2f_{max}\). In this case, the Nyquist rate is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). The question states that the signal is sampled at \(f_s = 25 \text{ kHz}\). Since \(25 \text{ kHz} < 30 \text{ kHz}\), the sampling rate is below the Nyquist rate. When a signal is undersampled (sampled below the Nyquist rate), aliasing occurs. Aliasing is the phenomenon where higher frequencies in the original signal are misinterpreted as lower frequencies in the sampled signal, leading to distortion and an inability to accurately reconstruct the original waveform. Therefore, the sampling process described will result in aliasing. The core concept tested here is the direct application of the Nyquist-Shannon sampling theorem. A sampling frequency of \(25 \text{ kHz}\) is insufficient to capture all the information in a signal with a maximum frequency of \(15 \text{ kHz}\). The theorem dictates that \(f_s > 2f_{max}\). Here, \(25 \text{ kHz} \ngtr 2 \times 15 \text{ kHz}\), or \(25 \text{ kHz} \ngtr 30 \text{ kHz}\). This violation of the theorem leads to aliasing. The explanation emphasizes that accurate reconstruction is impossible under these conditions, a crucial aspect of digital signal processing taught at institutions like ESIEE Paris, which focuses on applied sciences and engineering. Understanding aliasing is vital for designing effective digital systems, from audio processing to telecommunications, ensuring data integrity and signal fidelity. The ability to identify undersampling and its consequences is a foundational skill for any aspiring engineer in these fields.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning 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, i.e., \(f_s \ge 2f_{max}\). This minimum sampling rate is known as the Nyquist rate. Consider a scenario where a continuous-time signal, \(x(t)\), contains frequency components up to 15 kHz. According to the Nyquist-Shannon sampling theorem, the minimum sampling frequency required to avoid aliasing and allow for perfect reconstruction is twice the maximum frequency component. Therefore, the minimum sampling frequency \(f_{s,min}\) would be \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). If the signal is sampled at a frequency of 25 kHz, which is less than the Nyquist rate of 30 kHz, aliasing will occur. Aliasing is the phenomenon where higher frequencies in the original signal are incorrectly represented as lower frequencies in the sampled signal. This distortion makes it impossible to accurately reconstruct the original signal from the samples. The frequencies above \(f_s/2 = 25 \text{ kHz} / 2 = 12.5 \text{ kHz}\) will be folded back into the lower frequency range, corrupting the signal information. Therefore, the most accurate statement regarding the sampling of this signal at 25 kHz is that aliasing will occur, and the original signal cannot be perfectly reconstructed. This concept is foundational in digital signal processing, a core area of study within ESIEE Paris’s engineering programs, emphasizing the critical relationship between analog and digital domains. Understanding aliasing is crucial for designing effective analog-to-digital converters and ensuring the integrity of digital representations of real-world signals. The ability to identify situations where reconstruction is compromised due to undersampling is a key skill for ESIEE Paris students.

The core principle tested here is the understanding of how different communication protocols and their underlying mechanisms impact data integrity and efficiency in a networked environment, a fundamental concept for ESIEE Paris’s engineering programs. The scenario describes a distributed system where nodes need to share state information. Consider a scenario where a distributed system relies on a consensus mechanism for state synchronization. If the system uses a protocol that guarantees eventual consistency but does not enforce strict ordering or provide mechanisms for conflict resolution beyond simple last-writer-wins, then a situation where multiple nodes simultaneously update the same data item, followed by a period of network partition, would lead to divergent states. Upon network restoration, if the conflict resolution is merely based on the timestamp of the last received update, the system might converge to a state that doesn’t reflect the true aggregate intent of the operations. For instance, if Node A updates a counter to 5 at time \(t_1\) and Node B updates the same counter to 7 at time \(t_2\), and both updates are acknowledged locally before a partition. During the partition, Node A receives another update making the counter 6, and Node B receives an update making it 8. If the partition resolves and Node B’s update (7) arrives at Node A after Node A’s update (6), and Node A’s update (6) arrives at Node B after Node B’s update (8), and the resolution is strictly last-write-wins based on arrival time at the *receiving* node, the final state could be inconsistent. However, a more robust approach, particularly for critical state synchronization as emphasized in ESIEE Paris’s curriculum, would involve protocols that either prevent concurrent writes through locking mechanisms, use vector clocks for causal ordering, or employ more sophisticated conflict-free replicated data types (CRDTs) that can deterministically merge concurrent updates. The question probes the understanding of these trade-offs. The most appropriate response highlights the limitations of simple, non-ordered, or weakly ordered protocols in maintaining data integrity under concurrent modifications and network instability, which is a key consideration in designing reliable distributed systems taught at ESIEE Paris.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its implications for aliasing. 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. Consider a scenario where a continuous-time signal contains frequency components up to \(15 \text{ kHz}\). According to the Nyquist-Shannon sampling theorem, the minimum sampling frequency required to avoid aliasing and ensure perfect reconstruction is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). If the signal is sampled at \(25 \text{ kHz}\), which is below the Nyquist rate, aliasing will occur. Aliasing is the phenomenon where higher frequencies in the original signal are misrepresented as lower frequencies in the sampled signal, leading to distortion. Specifically, a frequency \(f\) sampled below the Nyquist rate will appear as \(|f – k f_s|\) for some integer \(k\), where \(f_s\) is the sampling frequency. For a frequency of \(15 \text{ kHz}\) sampled at \(25 \text{ kHz}\), the aliased frequency would be \(|15 \text{ kHz} – 1 \times 25 \text{ kHz}| = |-10 \text{ kHz}| = 10 \text{ kHz}\). Therefore, the highest frequency component of \(15 \text{ kHz}\) will be incorrectly represented as \(10 \text{ kHz}\) in the sampled data. This demonstrates a critical concept in digital signal processing taught at ESIEE Paris, emphasizing the importance of proper sampling for signal integrity, a foundational element in fields like telecommunications and audio processing. Understanding aliasing is crucial for designing effective anti-aliasing filters and selecting appropriate sampling rates in any digital system.

The core concept tested here is the understanding of **digital signal processing** and the **Nyquist-Shannon sampling theorem**, fundamental to fields like telecommunications and embedded systems, both key areas at ESIEE Paris. The theorem states that to perfectly reconstruct a signal, the sampling frequency must be at least twice the highest frequency component of the signal. Let the original analog signal have a maximum frequency component \(f_{max}\). According to the Nyquist-Shannon sampling theorem, the minimum sampling frequency \(f_s\) required for perfect reconstruction is \(f_s \ge 2 \cdot f_{max}\). In this scenario, the analog signal is passed through a low-pass filter with a cutoff frequency of \(f_{cutoff} = 5 \text{ kHz}\). This filter removes any frequency components above \(5 \text{ kHz}\). Therefore, the maximum frequency component present in the signal *after* filtering is \(f_{max} = 5 \text{ kHz}\). Applying the Nyquist-Shannon sampling theorem, the minimum sampling frequency required for perfect reconstruction of this filtered signal is: \(f_{s, min} = 2 \cdot f_{max}\) \(f_{s, min} = 2 \cdot 5 \text{ kHz}\) \(f_{s, min} = 10 \text{ kHz}\) Any sampling frequency below \(10 \text{ kHz}\) would lead to aliasing, where higher frequencies masquerade as lower frequencies, distorting the reconstructed signal. Therefore, a sampling frequency of \(10 \text{ kHz}\) is the theoretical minimum to avoid information loss. This question probes the candidate’s grasp of how signal processing techniques, like filtering, interact with sampling requirements, a crucial aspect of digital signal processing and its applications in areas like sensor data acquisition and communication systems taught at ESIEE Paris. Understanding the implications of sampling rates is vital for designing efficient and accurate digital systems, preventing data corruption, and ensuring the integrity of transmitted or processed information. It highlights the practical application of theoretical principles in engineering disciplines.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its implications for aliasing. The Nyquist-Shannon sampling 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., \(f_s \ge 2f_{max}\). Aliasing occurs when this condition is violated, causing high-frequency components to be misrepresented as lower frequencies. In this scenario, the original analog signal contains frequencies up to 15 kHz. Therefore, the minimum sampling frequency required to avoid aliasing is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). The system samples this signal at 25 kHz. Since \(25 \text{ kHz} < 30 \text{ kHz}\), aliasing will occur. When aliasing occurs, a frequency \(f\) greater than \(f_s/2\) will be aliased to a frequency \(f_{alias}\) within the range \([0, f_s/2]\). The aliased frequency can be calculated using the formula \(f_{alias} = |f – k \cdot f_s|\), where \(k\) is an integer chosen such that \(0 \le f_{alias} < f_s/2\). The folding frequency is \(f_s/2\). The highest frequency component in the original signal is 15 kHz. The sampling frequency is 25 kHz. The folding frequency is \(25 \text{ kHz} / 2 = 12.5 \text{ kHz}\). Since 15 kHz is greater than the folding frequency of 12.5 kHz, it will be aliased. To find the aliased frequency of 15 kHz, we use the formula with \(f_s = 25 \text{ kHz}\): For \(k=0\), \(f_{alias} = |15 \text{ kHz} – 0 \cdot 25 \text{ kHz}| = 15 \text{ kHz}\). This is not less than \(f_s/2\). For \(k=1\), \(f_{alias} = |15 \text{ kHz} – 1 \cdot 25 \text{ kHz}| = |-10 \text{ kHz}| = 10 \text{ kHz}\). Since \(10 \text{ kHz} < 12.5 \text{ kHz}\), the aliased frequency is 10 kHz. This understanding is crucial for students at ESIEE Paris, particularly in programs related to signal processing, telecommunications, and embedded systems, where accurate digital representation of analog signals is paramount. Failing to adhere to sampling theorem principles can lead to significant distortion and misinterpretation of data, impacting the performance and reliability of electronic systems. The ability to predict and mitigate aliasing is a core competency for engineers in these fields, ensuring the integrity of information transmitted and processed.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning 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, i.e., \(f_s \ge 2f_{max}\). This minimum sampling rate is known as the Nyquist rate. Consider a scenario where a continuous-time signal has its highest frequency component at \(f_{max} = 5 \text{ kHz}\). According to the Nyquist-Shannon sampling theorem, the minimum sampling frequency required to avoid aliasing and ensure perfect reconstruction is \(f_{s,min} = 2 \times f_{max}\). Calculation: \(f_{s,min} = 2 \times 5 \text{ kHz} = 10 \text{ kHz}\) Therefore, a sampling frequency of 10 kHz is the absolute minimum required. Any sampling frequency below this threshold would lead to aliasing, where higher frequencies in the original signal masquerade as lower frequencies in the sampled signal, making accurate reconstruction impossible. Conversely, sampling at a frequency higher than the Nyquist rate (e.g., 12 kHz or 15 kHz) would still allow for perfect reconstruction, although it might lead to increased data storage and processing requirements. The core concept tested is the direct relationship between the signal’s bandwidth and the necessary sampling rate for faithful representation, a cornerstone of digital signal processing taught at institutions like ESIEE Paris. This understanding is crucial for designing effective analog-to-digital converters and subsequent digital signal processing algorithms, ensuring the integrity of information captured from the physical world.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its implications for signal reconstruction. 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 rate is known as the Nyquist rate, \(f_{Nyquist} = 2 \times f_{max}\). In this case, \(f_{max} = 15 \text{ kHz}\). Therefore, the Nyquist rate is \(f_{Nyquist} = 2 \times 15 \text{ kHz} = 30 \text{ kHz}\). The question asks about the consequence of sampling this signal at \(f_s = 25 \text{ kHz}\). Since \(f_s = 25 \text{ kHz}\) is less than the Nyquist rate of \(30 \text{ kHz}\), the sampling is sub-Nyquist. When a signal is sampled below its Nyquist rate, aliasing occurs. Aliasing is an effect where higher frequencies in the original signal are incorrectly represented as lower frequencies in the sampled signal. This distortion makes it impossible to perfectly reconstruct the original continuous-time signal from the samples. The specific frequencies that fold back into the baseband (from 0 to \(f_s/2\)) are those above \(f_s/2\). In this scenario, frequencies above \(25 \text{ kHz} / 2 = 12.5 \text{ kHz}\) will be aliased. Since the original signal contains frequencies up to \(15 \text{ kHz}\), these frequencies, being greater than \(12.5 \text{ kHz}\), will be aliased. This aliasing corrupts the information in the sampled signal, preventing accurate reconstruction of the original \(x(t)\). The fundamental concept being tested is the direct application of the Nyquist criterion and the understanding of the phenomenon of aliasing when this criterion is violated. This is a core concept in signal processing, crucial for students at ESIEE Paris, especially in programs related to electronics, telecommunications, and computer science, where digital signal manipulation is prevalent. Understanding aliasing is vital for designing effective sampling strategies and for interpreting the results of digital signal processing.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its implications for aliasing. The Nyquist-Shannon sampling 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., \(f_s \ge 2f_{max}\). Aliasing occurs when this condition is violated, meaning the sampling frequency is less than twice the maximum frequency (\(f_s < 2f_{max}\)). In such a case, higher frequencies in the analog signal are misrepresented as lower frequencies in the sampled digital signal, leading to distortion. Consider a scenario where an analog signal contains frequency components up to \(f_{max} = 15\) kHz. If this signal is sampled at a rate of \(f_s = 25\) kHz, we can determine if aliasing will occur. According to the Nyquist-Shannon theorem, the minimum required sampling rate to avoid aliasing is \(2 \times f_{max} = 2 \times 15 \text{ kHz} = 30 \text{ kHz}\). Since the actual sampling rate \(f_s = 25\) kHz is less than the required minimum of 30 kHz (\(25 \text{ kHz} < 30 \text{ kHz}\)), aliasing will occur. The frequencies above \(f_s/2 = 25 \text{ kHz} / 2 = 12.5 \text{ kHz}\) will be aliased. Specifically, a frequency component at \(f\) where \(12.5 \text{ kHz} < f \le 15 \text{ kHz}\) will appear as \(|f – k \cdot f_s|\) for some integer \(k\), such that the aliased frequency is within the range \([0, f_s/2]\). For instance, a frequency of 14 kHz would alias to \(|14 \text{ kHz} – 1 \cdot 25 \text{ kHz}| = |-11 \text{ kHz}| = 11 \text{ kHz}\), which is below the Nyquist frequency of 12.5 kHz. This phenomenon distorts the original signal's spectral content, making accurate reconstruction impossible without prior filtering. Therefore, the critical factor for preventing aliasing in this context is ensuring the sampling frequency is at least twice the maximum frequency of the signal.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its implications for aliasing. 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, i.e., \(f_s \ge 2f_{max}\). This minimum sampling rate is known as the Nyquist rate. In the given scenario, a continuous-time signal with a maximum frequency component of 15 kHz is being sampled. To avoid aliasing, the sampling frequency must be at least \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). If the sampling frequency is set to 25 kHz, which is less than the Nyquist rate of 30 kHz, aliasing will occur. Aliasing is the phenomenon where higher frequencies in the original signal are misrepresented as lower frequencies in the sampled signal, leading to distortion and loss of information. The specific frequency that appears as 5 kHz after sampling at 25 kHz would be a frequency \(f\) such that \(f \pmod{f_s} = 5 \text{ kHz}\) or \(f_s – (f \pmod{f_s}) = 5 \text{ kHz}\). For a frequency above \(f_s/2\), the aliased frequency is \(|f – k \cdot f_s|\) where \(k\) is an integer chosen such that the result is in the range \([0, f_s/2]\). A frequency of 25 kHz would alias to 0 kHz. A frequency of 20 kHz would alias to 20 kHz. A frequency of 30 kHz would alias to 5 kHz because \(30 \text{ kHz} – 25 \text{ kHz} = 5 \text{ kHz}\), and 5 kHz is within the range \([0, 25 \text{ kHz}/2]\). Therefore, a signal component at 30 kHz would be incorrectly interpreted as 5 kHz. This demonstrates a critical concept in signal processing relevant to ESIEE Paris’s curriculum in electrical engineering and computer science, where understanding signal integrity and data acquisition is paramount.

The question probes the understanding of the fundamental principles of digital signal processing, specifically concerning aliasing and its mitigation. Aliasing occurs when the sampling rate of a signal is insufficient to accurately represent its highest frequency components. According to the Nyquist-Shannon sampling theorem, the sampling frequency (\(f_s\)) must be at least twice the highest frequency (\(f_{max}\)) present in the signal to avoid aliasing, meaning \(f_s \ge 2f_{max}\). If a signal contains frequencies above \(f_s/2\), these higher frequencies will be misrepresented as lower frequencies within the sampled data, leading to distortion. In the given scenario, a continuous-time signal with a maximum frequency of 15 kHz is sampled at a rate of 20 kHz. The Nyquist frequency for this sampling rate is \(f_s/2 = 20 \text{ kHz} / 2 = 10 \text{ kHz}\). Since the signal contains frequencies up to 15 kHz, which is greater than the Nyquist frequency of 10 kHz, aliasing will occur. Specifically, the frequency component at 15 kHz will be aliased. The aliased frequency (\(f_{alias}\)) can be found by considering the difference between the original frequency and the nearest multiple of the sampling frequency. In this case, the 15 kHz component will fold back into the frequency band below the Nyquist frequency. The aliased frequency is calculated as \(|15 \text{ kHz} – k \cdot 20 \text{ kHz}|\), where \(k\) is an integer chosen such that the result is within the range \([0, f_s/2]\). For \(k=1\), \(|15 \text{ kHz} – 1 \cdot 20 \text{ kHz}| = |-5 \text{ kHz}| = 5 \text{ kHz}\). This 5 kHz is within the range \([0, 10 \text{ kHz}]\). Therefore, the 15 kHz component will appear as a 5 kHz component in the sampled signal. To prevent this, an anti-aliasing filter, which is a low-pass filter, should be applied *before* sampling. This filter removes or significantly attenuates frequencies in the original signal that are above the Nyquist frequency (\(f_s/2\)). In this case, the anti-aliasing filter should have a cutoff frequency below 15 kHz but above the highest desired frequency component to be preserved, ideally set at or below the Nyquist frequency of 10 kHz to ensure all components above it are removed before sampling. This process ensures that the sampled signal accurately represents the original signal’s frequency content up to the Nyquist frequency, thereby avoiding the distortion caused by aliasing. The correct approach is to use an anti-aliasing filter with a cutoff frequency at or below the Nyquist frequency before sampling.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its implications for aliasing. The scenario describes an analog signal \(x(t)\) with a maximum frequency component of \(f_{max} = 15 \text{ kHz}\). The Nyquist-Shannon sampling theorem states that 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, i.e., \(f_s \ge 2 f_{max}\). This minimum sampling frequency is known as the Nyquist rate. In this case, the Nyquist rate for the signal \(x(t)\) is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). The question states that the signal is sampled at \(f_s = 25 \text{ kHz}\). Since \(25 \text{ kHz} < 30 \text{ kHz}\), the sampling rate is below the Nyquist rate. When a signal is sampled below its Nyquist rate, higher frequency components in the original analog signal "fold over" or alias into lower frequencies in the sampled digital signal. Specifically, frequencies above \(f_s/2\) will appear as frequencies below \(f_s/2\). The highest frequency component in the original signal is \(15 \text{ kHz}\). The folding frequency, or Nyquist frequency, 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 alias. The frequency \(15 \text{ kHz}\) is above \(12.5 \text{ kHz}\). The aliased frequency \(f_{alias}\) can be calculated using the formula \(f_{alias} = |f – k \cdot f_s|\), where \(f\) is the original frequency and \(k\) is an integer chosen such that \(0 \le f_{alias} \le f_s/2\). For \(f = 15 \text{ kHz}\) and \(f_s = 25 \text{ kHz}\): We need to find an integer \(k\) such that \(0 \le |15 \text{ kHz} – k \cdot 25 \text{ kHz}| \le 12.5 \text{ kHz}\). If \(k=1\), \(|15 \text{ kHz} – 1 \cdot 25 \text{ kHz}| = |-10 \text{ kHz}| = 10 \text{ kHz}\). Since \(10 \text{ kHz} \le 12.5 \text{ kHz}\), the aliased frequency is \(10 \text{ kHz}\). Therefore, the \(15 \text{ kHz}\) component of the original analog signal will appear as a \(10 \text{ kHz}\) component in the sampled digital signal due to aliasing. This phenomenon is a critical consideration in the design of analog-to-digital converters (ADCs) and the selection of appropriate sampling rates to preserve signal integrity, a core concept taught in signal processing courses at ESIEE Paris. Understanding aliasing is crucial for avoiding distortion and ensuring accurate data representation in various engineering applications, from telecommunications to sensor data acquisition.

The core concept here revolves around the principles of electromagnetic induction and Faraday’s Law, particularly as applied to a rotating coil in a magnetic field, which is fundamental to the operation of AC generators, a key area of study in electrical engineering programs like those at ESIEE Paris. The induced electromotive force (EMF) in a coil rotating in a uniform magnetic field is given by the formula \( \mathcal{E} = NAB\omega \sin(\omega t) \), where \( N \) is the number of turns, \( A \) is the area of the coil, \( B \) is the magnetic field strength, and \( \omega \) is the angular velocity. The question asks about the conditions that maximize the *rate of change* of magnetic flux, which is directly proportional to the induced EMF. The magnetic flux through the coil is given by \( \Phi = NBA \cos(\omega t) \). The rate of change of flux is \( \frac{d\Phi}{dt} = -NBA\omega \sin(\omega t) \). The magnitude of this rate of change is maximized when \( |\sin(\omega t)| = 1 \), which occurs when \( \omega t = \frac{\pi}{2} + n\pi \) for any integer \( n \). At these moments, the angle between the magnetic field vector and the normal to the plane of the coil is \( \frac{\pi}{2} \) radians (or 90 degrees). This means the plane of the coil is parallel to the magnetic field lines. In this orientation, the flux through the coil is momentarily zero, but its rate of change is at its peak. Therefore, the maximum rate of change of magnetic flux, and consequently the maximum induced EMF, occurs when the coil’s plane is parallel to the magnetic field.

The scenario describes a system where a sensor network is deployed to monitor environmental conditions, specifically air quality, across a campus. The core challenge is to ensure that the data collected by these sensors is both accurate and timely, given potential network disruptions. The question probes the understanding of network reliability and data integrity in a distributed sensing environment, which is a key consideration in fields like embedded systems and data science, areas of focus at ESIEE Paris. The concept of redundancy in network design is crucial here. If a single point of failure exists, the entire system’s reliability is compromised. Implementing multiple communication paths for data transmission from the sensors to the central processing unit mitigates this risk. If one path is disrupted (e.g., due to interference, hardware failure, or physical obstruction), data can still be routed through an alternative path. This ensures continuous data flow and prevents data loss. Furthermore, the explanation of data integrity involves mechanisms to verify that the data received is the same as the data transmitted. This can be achieved through error detection and correction codes. However, the primary concern in this scenario, given the potential for network disruption, is the *availability* of the data. Without a reliable path, even error-free data cannot be delivered. Therefore, the most effective strategy to address the potential for network disruptions and ensure continuous data acquisition for ESIEE Paris’s environmental monitoring project is to implement robust network redundancy. This involves designing the network with multiple, independent pathways for data transmission, ensuring that the failure of any single component or link does not lead to a complete loss of data. This principle of fault tolerance is fundamental in designing resilient systems, a core competency fostered at ESIEE Paris.

The core concept tested here is the understanding of signal-to-noise ratio (SNR) in the context of digital communication and its impact on data integrity. While no direct calculation is performed, the explanation implicitly relies on the definition of SNR as the ratio of signal power to noise power. A higher SNR indicates a clearer signal with less interference, leading to more reliable data transmission. In the context of ESIEE Paris’s engineering programs, particularly those focusing on telecommunications and signal processing, understanding how environmental factors affect SNR is crucial for designing robust communication systems. For instance, in a wireless communication scenario, factors like electromagnetic interference from other devices, atmospheric conditions, and the physical distance between transmitter and receiver all contribute to the noise floor. Conversely, signal strength is influenced by transmission power and antenna efficiency. Therefore, a system operating in an environment with significant electromagnetic pollution and a weak signal will inherently have a lower SNR. This lower SNR directly translates to a higher probability of bit errors, necessitating more sophisticated error correction codes or retransmission mechanisms, which in turn can reduce effective data throughput. The question probes the candidate’s ability to connect these fundamental principles to a practical engineering challenge.

The core concept tested here is the understanding of signal-to-noise ratio (SNR) in the context of digital communication systems, a fundamental topic in electrical engineering and computer science programs at ESIEE Paris. While no direct calculation is required, the explanation of the correct answer involves understanding how SNR impacts data integrity and the efficiency of error correction. Signal-to-Noise Ratio (SNR) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. A higher SNR indicates that the signal is stronger relative to the noise, leading to clearer reception and fewer errors. In digital communication, this translates directly to the probability of bit errors. Consider a scenario where a digital signal is transmitted. Noise, which can be introduced by various sources like thermal noise, interference, or quantization errors, corrupts the signal. The receiver attempts to decode the original digital bits (0s and 1s) from the received, noisy signal. If the signal is significantly degraded by noise (low SNR), it becomes difficult for the receiver to distinguish between a transmitted ‘0’ and a transmitted ‘1’, leading to incorrect decoding. The impact of SNR on error rates is generally inverse. As SNR increases, the probability of a bit error decreases. This is because a higher SNR means the signal amplitude is more distinct from the noise amplitude, making it easier for the receiver’s decision circuitry to correctly identify the intended bit. Conversely, a low SNR means the noise is comparable to or even greater than the signal, making accurate decoding highly improbable. Therefore, a system operating with a high SNR will exhibit a lower bit error rate (BER) compared to a system with a low SNR, assuming all other factors remain constant. This is a critical consideration in designing reliable communication links, as it directly affects the throughput and the need for retransmissions or sophisticated error correction codes. The ability to maintain a sufficient SNR is paramount for achieving the desired performance metrics in any digital communication system, a principle deeply embedded in the curriculum at ESIEE Paris.

The scenario describes a system where a signal is transmitted through a medium that introduces attenuation and noise. The goal is to determine the signal-to-noise ratio (SNR) at the receiver. First, we need to calculate the received signal power. The transmitted power is \(P_{tx} = 10 \text{ mW}\). The attenuation factor is \(A = 0.5\). Therefore, the received signal power is \(P_{rx} = P_{tx} \times A = 10 \text{ mW} \times 0.5 = 5 \text{ mW}\). Next, we need to determine the noise power at the receiver. The noise is characterized by its power spectral density \(N_0 = 2 \times 10^{-10} \text{ W/Hz}\) and the bandwidth of the receiver is \(B = 10 \text{ kHz}\). The total noise power is \(P_n = N_0 \times B = (2 \times 10^{-10} \text{ W/Hz}) \times (10 \times 10^3 \text{ Hz}) = 2 \times 10^{-6} \text{ W}\). Finally, the signal-to-noise ratio (SNR) is calculated as the ratio of received signal power to noise power: \(SNR = \frac{P_{rx}}{P_n} = \frac{5 \text{ mW}}{2 \times 10^{-6} \text{ W}} = \frac{5 \times 10^{-3} \text{ W}}{2 \times 10^{-6} \text{ W}} = 2.5 \times 10^3 = 2500\). The SNR is often expressed in decibels (dB). The conversion formula is \(SNR_{dB} = 10 \log_{10}(SNR)\). \(SNR_{dB} = 10 \log_{10}(2500) \approx 10 \times 3.3979 \approx 33.98 \text{ dB}\). This calculation is fundamental in communication systems engineering, a core discipline at ESIEE Paris. Understanding SNR is crucial for evaluating the quality of a communication link and determining the feasibility of transmitting information reliably over a noisy channel. A higher SNR indicates a stronger signal relative to the background noise, leading to fewer errors in data reception. The concepts of signal attenuation, noise power spectral density, bandwidth, and their impact on SNR are central to courses in signal processing, telecommunications, and embedded systems at ESIEE Paris. The ability to perform such calculations and interpret their meaning demonstrates a candidate’s foundational knowledge in electrical engineering and information technology, aligning with the rigorous academic standards of ESIEE Paris.

The scenario describes a system where a signal is transmitted through a medium with a specific attenuation factor. The initial signal strength is given as \(P_0 = 100\) units. After traversing a distance \(d\), the signal strength \(P(d)\) is reduced by a factor of \(k\) for every unit of distance. This implies an exponential decay model: \(P(d) = P_0 \cdot k^d\). The problem states that after \(5\) units of distance, the signal strength is \(P(5) = 20\) units. We can use this information to find the attenuation factor \(k\). \[P(5) = P_0 \cdot k^5\] \[20 = 100 \cdot k^5\] Divide both sides by 100: \[\frac{20}{100} = k^5\] \[0.2 = k^5\] To find \(k\), we take the fifth root of 0.2: \[k = (0.2)^{1/5}\] Now, we need to find the distance \(d_{crit}\) at which the signal strength drops to \(1\) unit. \[P(d_{crit}) = P_0 \cdot k^{d_{crit}}\] \[1 = 100 \cdot k^{d_{crit}}\] Divide both sides by 100: \[\frac{1}{100} = k^{d_{crit}}\] Substitute the value of \(k\): \[\frac{1}{100} = ((0.2)^{1/5})^{d_{crit}}\] \[\frac{1}{100} = (0.2)^{d_{crit}/5}\] To solve for \(d_{crit}\), we can take the logarithm of both sides. Using the natural logarithm (ln): \[\ln\left(\frac{1}{100}\right) = \ln\left((0.2)^{d_{crit}/5}\right)\] Using logarithm properties: \[\ln(1) – \ln(100) = \frac{d_{crit}}{5} \cdot \ln(0.2)\] Since \(\ln(1) = 0\): \[-\ln(100) = \frac{d_{crit}}{5} \cdot \ln(0.2)\] Rearrange to solve for \(d_{crit}\): \[d_{crit} = \frac{-5 \cdot \ln(100)}{\ln(0.2)}\] Now, we can calculate the numerical value. \(\ln(100) \approx 4.60517\) \(\ln(0.2) \approx -1.60944\) \[d_{crit} \approx \frac{-5 \cdot 4.60517}{-1.60944}\] \[d_{crit} \approx \frac{-23.02585}{-1.60944}\] \[d_{crit} \approx 14.3067\] The question asks for the critical distance at which the signal strength reduces to 1 unit. This is a fundamental concept in signal propagation and loss, relevant to telecommunications and sensor networks, areas of study at ESIEE Paris. Understanding exponential decay is crucial for designing efficient communication systems and predicting signal behavior over distance. The calculation demonstrates how to determine the range of effective signal transmission, a key consideration in engineering disciplines. The critical distance represents the maximum effective range before the signal becomes unusable due to attenuation. This involves applying logarithmic properties to solve for an unknown exponent, a common mathematical technique in scientific modeling.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning 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, i.e., \(f_s \ge 2f_{max}\). This minimum sampling frequency is known as the Nyquist rate. Consider a scenario where a continuous-time signal contains frequency components up to \(15\) kHz. To avoid aliasing, which is the distortion that occurs when the sampling frequency is too low, the sampling frequency must be greater than or equal to twice the maximum frequency. Therefore, the minimum required sampling frequency is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). If a signal is sampled at \(25 \text{ kHz}\), which is below the Nyquist rate of \(30 \text{ kHz}\), aliasing will occur. This means that frequencies above \(f_s/2 = 25 \text{ kHz} / 2 = 12.5 \text{ kHz}\) will be incorrectly represented as lower frequencies in the sampled data. Specifically, a frequency component at \(15 \text{ kHz}\) would be aliased to \(|15 \text{ kHz} – 25 \text{ kHz}| = |-10 \text{ kHz}| = 10 \text{ kHz}\). This phenomenon corrupts the original signal information, making accurate reconstruction impossible without prior knowledge or filtering. The core concept tested here is the direct application of the Nyquist criterion to determine the feasibility of signal reconstruction and to predict the consequences of undersampling. This is a foundational concept for students at ESIEE Paris, particularly in programs related to electronics, telecommunications, and signal processing, as it underpins the design of analog-to-digital converters and the integrity of digital communication systems. Understanding aliasing is crucial for designing effective anti-aliasing filters and selecting appropriate sampling rates to preserve signal fidelity.

The core of this question lies in understanding the fundamental principles of digital signal processing, specifically the Nyquist-Shannon sampling theorem and its implications for aliasing. 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. Mathematically, this is expressed as \(f_s \ge 2f_{max}\). In the given scenario, the analog signal contains frequency components up to 15 kHz. Therefore, the minimum sampling frequency required to avoid aliasing is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). The question states that the signal is sampled at 25 kHz. Since 25 kHz is less than the required minimum of 30 kHz, aliasing will occur. Aliasing causes higher frequencies in the original signal to be misrepresented as lower frequencies in the sampled signal. Specifically, frequencies above \(f_s/2\) (the Nyquist frequency) will be folded back into the frequency range below \(f_s/2\). The folding frequency for a given frequency \(f\) above \(f_s/2\) is given by \(|f – k \cdot f_s|\), where \(k\) is an integer chosen such that the result is within the range \([0, f_s/2]\). For the 15 kHz component, which is above the Nyquist frequency of \(25 \text{ kHz} / 2 = 12.5 \text{ kHz}\), it will be aliased. The closest multiple of the sampling frequency (25 kHz) to 15 kHz is 25 kHz itself. The aliased frequency will be \(|15 \text{ kHz} – 1 \times 25 \text{ kHz}| = |-10 \text{ kHz}| = 10 \text{ kHz}\). This means the 15 kHz component will appear as a 10 kHz signal in the sampled data. The question asks about the consequence of sampling at 25 kHz when the signal has components up to 15 kHz. The most direct and significant consequence is the introduction of aliasing, specifically the misrepresentation of the highest frequency component. Therefore, the 15 kHz component will be incorrectly represented as a lower frequency, which is 10 kHz. This phenomenon is a critical concept in digital signal processing taught at institutions like ESIEE Paris, emphasizing the importance of proper sampling rates for accurate signal reconstruction. Understanding aliasing is crucial for designing effective digital systems and avoiding data corruption, a key skill for future engineers.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its implications for aliasing. 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 to avoid aliasing is \(f_s \ge 2 \times 15 \text{ kHz} = 30 \text{ kHz}\). The question asks about the consequence of sampling at \(25 \text{ kHz}\). Since \(25 \text{ kHz} < 30 \text{ kHz}\), the sampling rate is below the Nyquist rate. When the sampling frequency is less than twice the maximum signal frequency, higher frequencies in the original signal are misrepresented as lower frequencies in the sampled signal. This phenomenon is known as aliasing. Specifically, frequencies above \(f_s/2\) will be aliased. In this case, \(f_s/2 = 25 \text{ kHz} / 2 = 12.5 \text{ kHz}\). Any frequency component in the original signal above \(12.5 \text{ kHz}\) will appear as a lower frequency within the range of \(0\) to \(12.5 \text{ kHz}\) after sampling. The frequencies between \(12.5 \text{ kHz}\) and \(15 \text{ kHz}\) will be aliased. For instance, a frequency \(f\) such that \(12.5 \text{ kHz} < f \le 15 \text{ kHz}\) will appear as \(|f – k \cdot f_s|\) for some integer \(k\), where \(|f – k \cdot f_s| \le f_s/2\). The most significant aliasing will occur for the highest frequency components. For example, a frequency of \(15 \text{ kHz}\) would be aliased to \(|15 \text{ kHz} – 1 \cdot 25 \text{ kHz}| = |-10 \text{ kHz}| = 10 \text{ kHz}\). This distortion means that the original signal's spectral content above \(12.5 \text{ kHz}\) cannot be accurately recovered from the sampled data. The introduction of these spurious lower frequencies corrupts the signal's fidelity and prevents faithful reconstruction. This is a critical concept in digital signal processing, relevant to fields like telecommunications and audio engineering, which are foundational to many programs at ESIEE Paris. Understanding aliasing is essential for designing effective anti-aliasing filters and selecting appropriate sampling rates to preserve signal integrity, a core skill for engineers graduating from ESIEE Paris.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its implications for aliasing. The Nyquist-Shannon sampling 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, i.e., \(f_s \ge 2f_{max}\). This minimum sampling frequency is known as the Nyquist rate. In this scenario, the analog signal has a maximum frequency component of 15 kHz. Therefore, according to the Nyquist-Shannon sampling theorem, the minimum sampling frequency required to avoid aliasing is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). The question asks about the consequence of sampling at a frequency *below* this critical rate. When a signal is sampled at a frequency \(f_s\) lower than the Nyquist rate, higher frequency components in the original signal are misrepresented as lower frequencies in the sampled signal. This phenomenon is called aliasing. Specifically, a frequency \(f\) in the original signal will appear as \(|f – k f_s|\) in the sampled signal, where \(k\) is an integer chosen such that the resulting frequency is within the range \([0, f_s/2]\). If the sampling frequency is 20 kHz, which is less than the required 30 kHz, then frequencies above \(f_s/2 = 20 \text{ kHz} / 2 = 10 \text{ kHz}\) will be aliased. The highest frequency component in the original signal is 15 kHz. This 15 kHz component, when sampled at 20 kHz, will be aliased. The aliased frequency can be calculated as \(|15 \text{ kHz} – 1 \times 20 \text{ kHz}| = |-5 \text{ kHz}| = 5 \text{ kHz}\). This means that the 15 kHz component will be indistinguishable from a 5 kHz component in the sampled data. This distortion, where higher frequencies masquerade as lower frequencies, is the fundamental problem of aliasing. The ability to correctly identify this phenomenon and its cause is crucial for any student pursuing studies in signal processing or related fields at ESIEE Paris, as it underpins the integrity of digital representations of analog phenomena. Understanding aliasing is essential for designing effective anti-aliasing filters and choosing appropriate sampling rates in various engineering applications, from telecommunications to audio and image processing, all of which are areas of focus within ESIEE Paris’s curriculum.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its implications for aliasing. 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, i.e., \(f_s \ge 2f_{max}\). This minimum sampling rate is known as the Nyquist rate. Consider a scenario where a continuous-time signal contains frequency components up to 15 kHz. According to the Nyquist-Shannon sampling theorem, the minimum sampling frequency required to avoid aliasing and ensure perfect reconstruction is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). If the signal is sampled at a frequency lower than this, such as 25 kHz, then frequencies above \(f_s/2 = 25 \text{ kHz} / 2 = 12.5 \text{ kHz}\) will be aliased. Specifically, a frequency component at 15 kHz, when sampled at 25 kHz, will appear as \(|15 \text{ kHz} – n \times 25 \text{ kHz}|\) for some integer \(n\). For \(n=1\), this is \(|15 \text{ kHz} – 25 \text{ kHz}| = |-10 \text{ kHz}| = 10 \text{ kHz}\). Thus, the 15 kHz component will be misrepresented as a 10 kHz component in the sampled data. This phenomenon of higher frequencies masquerading as lower frequencies due to insufficient sampling is aliasing. Therefore, sampling at 25 kHz when the signal’s maximum frequency is 15 kHz will lead to aliasing, as 25 kHz is less than the required Nyquist rate of 30 kHz. The consequence is that the original 15 kHz frequency will be incorrectly perceived as 10 kHz.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its implications for aliasing. The theorem states that to perfectly reconstruct a continuous-time signal from its samples, 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. Consider a signal containing frequency components up to \(f_{max} = 15\) kHz. According to the Nyquist-Shannon sampling theorem, the minimum sampling frequency required to avoid aliasing is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). If a signal is sampled at a frequency lower than this minimum, higher frequency components will “fold back” into the lower frequency range, appearing as lower frequencies that were not originally present. This phenomenon is called aliasing. The question asks about the consequence of sampling a signal with a maximum frequency of 15 kHz at a rate of 25 kHz. Since 25 kHz is less than the required Nyquist rate of 30 kHz, aliasing will occur. Specifically, frequencies above \(f_s/2 = 25 \text{ kHz}/2 = 12.5 \text{ kHz}\) will be aliased. A frequency component at 15 kHz, when sampled at 25 kHz, will appear as a lower frequency. The aliased frequency (\(f_{alias}\)) can be calculated using the formula \(f_{alias} = |f – n \cdot f_s|\), where \(f\) is the original frequency and \(n\) is an integer chosen such that \(0 \le f_{alias} < f_s/2\). For \(f = 15 \text{ kHz}\) and \(f_s = 25 \text{ kHz}\), we can find \(n\). If \(n=1\), \(f_{alias} = |15 \text{ kHz} – 1 \cdot 25 \text{ kHz}| = |-10 \text{ kHz}| = 10 \text{ kHz}\). Since 10 kHz is within the range \(0 \le f_{alias} < 12.5 \text{ kHz}\), this is the aliased frequency. Therefore, a 15 kHz component will be indistinguishable from a 10 kHz component. This understanding is crucial in digital signal processing, analog-to-digital conversion, and data acquisition systems, areas of study within ESIEE Paris's engineering programs, ensuring that signals are sampled appropriately to preserve their original information content without distortion.

The scenario describes a system where a signal is transmitted and received. The core concept being tested is the understanding of signal integrity and potential sources of degradation in a communication channel, particularly relevant to fields like telecommunications and embedded systems studied at ESIEE Paris. The signal starts as a clean digital pulse. When transmitted through a physical medium, it is subject to various forms of interference and distortion. Attenuation refers to the loss of signal strength over distance. Noise is any unwanted random fluctuation that corrupts the signal. Bandwidth limitation of the channel can cause signal dispersion, where different frequency components of the signal travel at different speeds, leading to pulse spreading. Reflections occur when there are impedance mismatches in the transmission line, causing parts of the signal to bounce back. Jitter is the temporal deviation of a signal’s edge from its ideal position. Considering the options: – **Increased signal-to-noise ratio (SNR)**: This would imply the signal is *less* affected by noise, which is contrary to the expected degradation. – **Reduced signal bandwidth**: While bandwidth limitations cause distortion, the statement “reduced signal bandwidth” itself doesn’t directly describe the *effect* on the signal’s waveform as much as the cause of some effects. More importantly, the question asks about the *primary* consequence of transmission through a non-ideal channel. – **Enhanced signal fidelity**: This is the opposite of what happens in a real-world transmission channel; fidelity is reduced. – **Temporal dispersion of the digital pulse**: This is a direct consequence of the signal’s constituent frequencies experiencing different delays within the channel, often due to bandwidth limitations or material properties of the transmission medium. This spreading of the pulse is a critical factor in determining the maximum data rate and the need for equalization techniques, which are fundamental topics in signal processing and communication systems at ESIEE Paris. The pulse spreading can lead to inter-symbol interference (ISI) if the spread of one pulse overlaps with the next. Therefore, temporal dispersion is the most accurate description of a primary degradation effect on the digital pulse waveform itself as it traverses a real-world communication channel.

The core of this question lies in understanding the principles of electromagnetic induction and Lenz’s Law, which are fundamental to many ESIEE Paris engineering disciplines, particularly in electrical engineering and mechatronics. 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. Consider a scenario where a permanent magnet is dropped through a non-magnetic, electrically conductive tube (like copper or aluminum). As the magnet falls, its magnetic field lines pass through the material of the tube. According to Faraday’s Law of Induction, a changing magnetic flux through a conductor induces an electromotive force (EMF), which in turn drives an induced current. These induced currents, known as eddy currents, flow in loops within the conductive tube. Lenz’s Law dictates the direction of these eddy currents. As the magnet moves downwards, the magnetic flux through the tube’s cross-section is changing. The eddy currents induced will create their own magnetic field that opposes this change. Specifically, as the magnet approaches a section of the tube, the eddy currents will generate a magnetic field that repels the approaching pole of the magnet. Conversely, as the magnet moves away from a section, the eddy currents will generate a magnetic field that attracts the receding pole of the magnet. This continuous opposition to the magnet’s motion results in a significant braking effect, causing the magnet to fall much slower than it would in free fall. The key is that the tube is non-magnetic. If the tube were magnetic, the interaction would be more complex, involving both induced currents and the inherent magnetic properties of the tube material. The absence of magnetic properties in the tube isolates the effect to electromagnetic induction and Lenz’s Law. Therefore, the primary phenomenon responsible for the observed slow descent is the magnetic braking caused by eddy currents induced in the conductive tube, which oppose the magnet’s motion.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the aliasing phenomenon and its mitigation. Aliasing occurs when the sampling rate of a signal is insufficient to accurately represent its highest frequency components. The Nyquist-Shannon sampling theorem states that to perfectly reconstruct a signal, the sampling frequency (\(f_s\)) must be at least twice the highest frequency (\(f_{max}\)) present in the signal, i.e., \(f_s \ge 2f_{max}\). This minimum sampling frequency is known as the Nyquist rate. In the given scenario, a signal containing frequencies up to \(15 \text{ kHz}\) is sampled at \(25 \text{ kHz}\). To determine if aliasing will occur, we compare the sampling frequency to twice the maximum frequency. Here, \(f_{max} = 15 \text{ kHz}\). Therefore, the Nyquist rate required for this signal is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). Since the actual sampling frequency (\(25 \text{ kHz}\)) is less than the required Nyquist rate (\(30 \text{ kHz}\)), aliasing will occur. The frequencies above \(f_s/2\), which is \(25 \text{ kHz} / 2 = 12.5 \text{ kHz}\), will be folded back into the lower frequency band. Specifically, the component at \(15 \text{ kHz}\) will be aliased to \(25 \text{ kHz} – 15 \text{ kHz} = 10 \text{ kHz}\). To prevent aliasing, an anti-aliasing filter is used before sampling. This filter is a low-pass filter designed to attenuate or remove frequencies above \(f_s/2\). In this case, the anti-aliasing filter should have a cutoff frequency at or below \(12.5 \text{ kHz}\) to ensure that no frequencies higher than this are present in the signal when it is sampled at \(25 \text{ kHz}\). Therefore, the most appropriate cutoff frequency for the anti-aliasing filter is \(12.5 \text{ kHz}\).

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning aliasing and sampling. Aliasing occurs when the sampling frequency is not sufficiently high to accurately represent the original analog signal. The Nyquist-Shannon sampling theorem states that to perfectly reconstruct an analog signal from its samples, the sampling frequency (\(f_s\)) must be at least twice the highest frequency component (\(f_{max}\)) present in the signal, i.e., \(f_s \ge 2f_{max}\). In this scenario, the analog signal contains frequency components up to 15 kHz. Therefore, the minimum sampling frequency required to avoid aliasing is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). The analog-to-digital converter (ADC) is operating at a sampling rate of 25 kHz. Since 25 kHz is less than the required 30 kHz, aliasing will occur. When aliasing occurs, frequencies above \(f_s/2\) (the Nyquist frequency) are folded back into the lower frequency range. The folding occurs symmetrically around the Nyquist frequency. A frequency \(f\) where \(f > f_s/2\) will appear as \(|f – k \cdot f_s|\) for some integer \(k\) such that the result is within the range \([0, f_s/2]\). For a frequency of 12 kHz, which is greater than the Nyquist frequency of \(25 \text{ kHz} / 2 = 12.5 \text{ kHz}\), the aliased frequency will be \(|12 \text{ kHz} – 1 \cdot 25 \text{ kHz}| = |-13 \text{ kHz}| = 13 \text{ kHz}\). However, this calculation is incorrect as 12 kHz is *below* the Nyquist frequency of 12.5 kHz. Let’s re-evaluate. The signal has components up to 15 kHz. The sampling frequency is 25 kHz. The Nyquist frequency is \(f_s/2 = 25 \text{ kHz} / 2 = 12.5 \text{ kHz}\). Any frequency component in the original signal that is above 12.5 kHz will be aliased. The highest frequency component is 15 kHz. The aliased frequency for 15 kHz is calculated as: \(|15 \text{ kHz} – 1 \cdot 25 \text{ kHz}| = |-10 \text{ kHz}| = 10 \text{ kHz}\). This 10 kHz is the frequency that the 15 kHz component will appear as after sampling at 25 kHz. The question asks which frequency component *will be present* in the sampled signal due to aliasing. Since the original signal has components up to 15 kHz, and the sampling frequency is 25 kHz, the 15 kHz component will be aliased. The aliased frequency is \(|15 \text{ kHz} – 25 \text{ kHz}| = 10 \text{ kHz}\). Therefore, a 10 kHz component will be present in the sampled signal due to aliasing. This demonstrates a critical concept in digital signal processing taught at ESIEE Paris, emphasizing the importance of adhering to sampling theorem for accurate signal representation. Understanding aliasing is crucial for fields like telecommunications and sensor data processing, both areas of focus at ESIEE Paris.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the sampling theorem and its implications for signal reconstruction. The scenario describes a continuous-time signal \(x(t)\) with a maximum frequency component of \(f_{max} = 1500\) Hz. According to the Nyquist-Shannon sampling theorem, to perfectly reconstruct a band-limited continuous-time signal from its discrete samples, the sampling frequency \(f_s\) must be strictly greater than twice the maximum frequency component of the signal. This minimum sampling rate is known as the Nyquist rate, which is \(2f_{max}\). In this case, the Nyquist rate is \(2 \times 1500 \text{ Hz} = 3000 \text{ Hz}\). The question states that the signal is sampled at \(f_s = 2500\) Hz. Since \(2500 \text{ Hz} < 3000 \text{ Hz}\), the sampling rate is below the Nyquist rate. When a signal is undersampled (sampled below the Nyquist rate), aliasing occurs. Aliasing is the phenomenon where higher frequencies in the original signal are incorrectly represented as lower frequencies in the sampled signal. This distortion makes it impossible to accurately reconstruct the original continuous-time signal from its samples. Therefore, the primary consequence of sampling at 2500 Hz is the introduction of aliasing, preventing faithful reconstruction. The options provided test the understanding of this concept. Option a) correctly identifies aliasing as the consequence. Option b) suggests that the signal can be perfectly reconstructed, which is false due to undersampling. Option c) proposes that only frequencies above 1250 Hz are affected, which is an incomplete understanding of aliasing; it affects the entire spectrum in a complex way, folding higher frequencies into lower ones. Option d) implies that no information is lost, which is also incorrect as aliasing fundamentally corrupts the signal representation. The core principle tested here is the direct application of the Nyquist-Shannon sampling theorem and the understanding of its violation.

The core principle tested here is the understanding of signal-to-noise ratio (SNR) in the context of digital communication and its impact on data integrity. While no direct calculation is performed, the conceptual understanding of how noise affects signal clarity is paramount. A higher SNR indicates that the signal’s power is significantly greater than the background noise power, leading to more reliable data transmission. Conversely, a lower SNR means noise is more dominant, increasing the likelihood of errors. In the context of ESIEE Paris’s focus on engineering and information technology, understanding how to maximize SNR is crucial for designing robust communication systems, efficient data processing, and reliable sensor networks. Factors that degrade SNR include electromagnetic interference, thermal noise within components, and signal attenuation over distance. Techniques to improve SNR involve filtering, amplification, error correction coding, and careful system design to minimize noise sources. Therefore, a system operating with a high SNR is inherently more resilient to disturbances and can achieve higher data throughput with fewer errors, a fundamental consideration in fields like telecommunications, embedded systems, and signal processing, all areas of study at ESIEE Paris. The question probes the candidate’s ability to connect a fundamental signal processing metric to practical system performance and design considerations.