Signals And Systems Quiz -1 2018-19

A CD contains a finite and countable set of discrete information, as it stores data in the form of digital bits. Each bit can only have two possible values, 0 or 1, making it a discrete set. The trajectory of the Sun and the movement of water through a pipe are continuous systems, as they involve continuous variables and can have an infinite number of possible values. The universe time scale is also a continuous concept, as time is considered to be continuous and infinite. Therefore, the data on a CD is the only option that represents a discrete set of information/system.

An LTI system is said to be initially relaxed when the system's output is zero for a zero input. This means that even if there is no input signal applied to the system, the output remains at zero. This indicates that the system is not affected by any initial conditions or past inputs, and it is in a state of rest or equilibrium.

The statement is true because a unit step function is commonly used in mathematics and engineering to represent the act of closing a switch in a constant voltage system. The unit step function is defined as 0 for negative values and 1 for positive values, which accurately represents the sudden change in voltage when a switch is closed. Therefore, the statement is correct.

The principle of superposition states that the response of a linear system to multiple inputs can be determined by adding the individual responses of each input. This means that the principle is only applicable to linear systems, as nonlinear systems do not exhibit this additive property. Therefore, the correct answer is "linear systems only."

The transfer function is a mathematical representation that relates the output of a system to its input. It is commonly used in control systems engineering to analyze and design systems. The transfer function provides information about the system's behavior, such as its stability, frequency response, and transient response. It is a valuable tool for understanding and predicting the system's output based on its input.

Real-time instruments like oscilloscopes should be time invariant because they are designed to accurately measure and display signals in real-time. Time invariance means that the instrument's performance and measurements remain consistent regardless of the time at which they are taken. This is crucial for ensuring accurate and reliable measurements, especially when dealing with time-sensitive signals. Therefore, real-time instruments like oscilloscopes should be time invariant to maintain their effectiveness and precision.

A signal is considered to have finite energy when the total energy of the signal is finite or bounded. This means that the signal's energy does not increase or decrease infinitely over time. Signals with finite energy have a limited amount of energy content and are typically used in applications such as digital communication systems. Infinite energy signals, on the other hand, have an unbounded energy content and are not commonly encountered in practical applications. Zero energy signals have no energy content at all. Therefore, the correct answer is finite energy.

The property of the delta function that indicates the equality between the area under the product of a function with a shifted impulse and the value of the function located at the unit impulse instant is called "Sampling." Sampling refers to the process of capturing or measuring the value of a continuous function at discrete points in time or space. In this context, the delta function acts as a sampling function, allowing us to extract the value of the function at a specific point. Therefore, the correct answer is Sampling.

A time invariant system is a system that does not change its behavior over time. Therefore, regardless of any delay in the input, the output of a time invariant system will remain the same.

All of the options mentioned in the question can weaken a signal. Attenuation refers to the reduction in the strength of a signal as it travels through a medium or transmission path. Distortion occurs when the signal is altered or distorted during transmission, resulting in a loss of information. Noise refers to any unwanted interference or random fluctuations that can disrupt the signal. Therefore, all of these factors can contribute to the weakening of a signal.

The zero state response represents the system's response to its input when there are no initial conditions present. In other words, it only considers the effect of the input signal itself and disregards any initial energy stored in the system. This response is obtained by setting the initial conditions to zero and applying the input signal to the system. Therefore, the zero state response represents the system's output solely due to the input signal.

The integral of a unit step function is a ramp function of slope 1. This is because the unit step function represents a sudden jump from 0 to 1 at a certain point. Taking the integral of this function results in a linear increase over time, starting from 0 and increasing at a constant rate of 1. Therefore, the correct answer is a ramp function of slope 1.

A system is said to be casual if the output of the system depends on both the past and present inputs. This means that the current output is influenced not only by the current input but also by the previous inputs that have been fed into the system. The system takes into account the history of inputs in order to generate the output. Future inputs are not considered in a casual system because the output is determined by the inputs that have already occurred.

The number of samples present in an impulse response is referred to as its "length". This term is commonly used in signal processing to describe the size or duration of the impulse response, which represents the output of a system when stimulated with an impulse input. The length of the impulse response determines the level of detail and accuracy in the system's response to different input signals.

Damped sinusoids are sinusoid signals that are multiplied by decaying exponentials. This means that the amplitude of the sinusoid decreases over time due to the decaying exponential factor. The decaying exponential can be seen as a damping factor that reduces the magnitude of the sinusoid signal as time progresses. This is in contrast to growing exponentials, which would cause the amplitude of the sinusoid to increase over time.

An equalizer used to compensate distortion in a communication system aims to faithfully recover the original signal. An invertible system is capable of reversing the distortion introduced by the communication channel. Therefore, an equalizer that can accurately recover the original signal can be considered an illustration of an invertible system. This means that the correct answer is "Invertible system".

The unit delay block in a discrete time system requires memory to store the previous input. This block delays the input signal by one sampling period, allowing the system to remember the previous input value and use it in subsequent calculations. This memory is necessary for systems that require feedback or rely on past inputs to determine the current output. The adder and signal multiplier blocks do not require memory to store previous inputs, and the unit advance block is not commonly used in discrete time systems.

Memory is a necessary component for all causal systems because it allows the system to retain and recall past information or events. Without memory, the system would not be able to remember previous inputs and outputs, making it impossible to establish a cause-effect relationship. Memory enables the system to store and process information over time, which is essential for its functioning. Therefore, memory is a fundamental requirement for all causal systems.

The correct answer is x(t) = x(t +T0). This mathematical notation specifies the condition of periodicity for a continuous time signal. It states that the value of the signal at time t is equal to the value of the signal at time t + T0, where T0 is the period of the signal. This means that the signal repeats itself every T0 units of time.

The Nyquist sampling rate is determined by the highest frequency component in the signal. In this case, the highest frequency component is 300 pt. According to the Nyquist-Shannon sampling theorem, the sampling rate should be at least twice the highest frequency component. Therefore, the Nyquist sampling rate in this case would be 2 * 300 pt = 1/300.

Real-time systems concerned with the concept of causality are causal because they are designed to respond to events in a predictable and deterministic manner. Causality refers to the cause-and-effect relationship between events, where the occurrence of one event leads to the occurrence of another. In real-time systems, the order and timing of events are crucial, and the system's behavior is determined by the causal relationship between events. Therefore, real-time systems must be causal to ensure that events are processed in the correct order and that the system responds appropriately to changes in its environment.

The energy of a constant amplitude complex valued exponential sequence is infinite. This is because the energy of a sequence is calculated by summing the squared magnitudes of each sample. In the case of a complex exponential sequence, the magnitude of each sample is constant, resulting in an infinite sum when squared. Therefore, the energy of the sequence is infinite.

An initially relaxed system becomes unstable only if bounded input generates unbounded output. This means that even if the system starts in a relaxed state, it can become unstable if the input to the system is limited but the output produced by the system keeps increasing without any bounds. This indicates that the system is not able to handle the input properly and starts to exhibit unstable behavior.

Discrete systems are characterized by input and output quantized at certain levels. In these systems, the input and output values are represented by a finite set of discrete values or levels, rather than being continuous. This means that the system can only process and produce values that fall within these specific levels, without any intermediate values. Therefore, the correct answer is "discrete".

The waveform shown in the question is a multi valued digital signal. This can be determined by the presence of multiple distinct levels or values in the waveform, indicating that it is not a simple binary signal with only two possible values. Additionally, the waveform is not continuous like an analog signal, and it does not appear to be random noise. Therefore, the most appropriate explanation is that it is a multi valued digital signal.

A non-causal system is one in which the output at the present time depends on the input at a time instant in the future. This means that the system requires knowledge of future inputs in order to generate the current output. In a causal system, the output at the present time depends on the input at the current time or an earlier time. Therefore, the correct answer is that the output at the present depends on the input at a time instant in the future.

All of the given options are stable discrete time systems. Option 1 is stable because multiplying the input signal by a constant factor does not affect stability. Option 2 is stable because reversing the input signal does not affect stability. Option 3 is stable because adding a constant to the input signal does not affect stability. Option 4 is stable because taking the cosine of the input signal does not affect stability. Therefore, options 1, 2, 3, and 4 are all stable discrete time systems.

The function δ(t - b) represents an impulse function originating at t = b. This means that the function has a value of zero for all t except when t equals b, where it has an impulse or spike. The impulse function is often represented by a delta function, and in this case, the delta function is centered at t = b. Therefore, the correct answer is "an impulse function originating at t = b."

In the design of systems, the stability of a system is defined by whether a bounded input results in a bounded output. If a system is stable, it means that for all values of the input, the output will remain bounded. This ensures that the system will not exhibit any unpredictable or erratic behavior. Conversely, if a system is unstable, it means that there are some values of the input for which the output becomes unbounded, indicating that the system is not reliable or predictable. Therefore, the correct answer is that a system is stable if a bounded input gives a bounded output for all values of the input.

Continuous systems are characterized by inputs and outputs that can take any value within a particular set of values. This means that the variables in a continuous system can have infinite possible values within a given range. In contrast, analog systems have continuous inputs and outputs but are limited to a specific range of values, discrete systems have a finite number of possible values for inputs and outputs, and digital systems have discrete inputs and outputs that are represented by binary digits (0s and 1s). Therefore, continuous is the correct answer for this question.

The integration operation is not involved in the computation process of linear convolution. Linear convolution is a mathematical operation that combines two signals to produce a third signal. It involves folding, shifting, and multiplication operations. The folding operation reverses one of the signals, the shifting operation aligns the signals, and the multiplication operation multiplies the corresponding samples of the signals. However, the integration operation is not necessary for linear convolution and is therefore not associated with its computation process.

The term y(-1) indicates the initial condition of the system. This refers to the state of the system at time -1, or the state just before the input is applied. It represents the system's previous state and is important in understanding how the system will respond to the input. By considering the initial condition, we can analyze the behavior of the causal system and predict its response to the input.

The condition -1

A linear system is one that follows the rules of scaling and additivity. Scaling refers to the property where multiplying the input by a constant results in a corresponding multiplication of the output by the same constant. Additivity means that the system's response to the sum of two inputs is equal to the sum of the system's response to each individual input. Therefore, a linear system obeys both scaling and additivity rules.

The given system, y(n) = 3x[n] - 2x[n - 1], is a causal system because the output at any given time n only depends on the current input x[n] and the past input x[n - 1]. It does not depend on any future inputs. Therefore, it satisfies the causality property of a system.

A power signal is a signal that has a finite average power. This means that over a given period of time, the signal's power remains constant and does not approach infinity or zero. Power signals are commonly used in communication systems and can be measured and analyzed using techniques such as power spectral density. Signals with infinite average power would not be practical or feasible in most applications, while signals with zero average power would not transmit any useful information. Therefore, the correct answer is finite average power.

The given equation y(t) = x(t^2) represents a non-causal system because the output y(t) depends on the input x(t^2) at a future time t^2. It is linear because the equation does not involve any non-linear operations such as multiplication or exponentiation. It is time-variant because the output depends on the time variable t, which is squared in the equation. Therefore, the correct answer is non causal, linear and time-variant.

The integral of the product of the Dirac delta function and the sine function is equal to zero. This is because the Dirac delta function is zero everywhere except at t=0, where it is infinite. However, the integral of the sine function over any finite interval centered around t=0 is equal to zero, due to the symmetry of the sine function. Therefore, when multiplied by the Dirac delta function, the integral evaluates to zero.

A system is considered dynamic because its output is influenced by the past input. The system takes into account the previous input values to determine its current output. It does not rely on future input or present and future inputs alone to generate the output. Therefore, the past input is crucial in understanding the behavior and response of a dynamic system.

In a linear system, the output is directly proportional to the input. In this circuit, the behavior of the circuit is determined by the relationship between the current and voltage. If the initial voltage across the capacitor, v0, is zero, then the circuit will behave linearly. This means that the output voltage will be directly proportional to the input current. However, if v0 is non-zero, the circuit will exhibit non-linear behavior, as the output voltage will not be directly proportional to the input current. Therefore, the system is linear only if v0 is zero.

If the sequence y(n) is given by y(n) = x(-n), then it is causal. A causal system is one in which the output at any given time depends only on the present and past values of the input. In this case, the output y(n) depends on the past values of the input x(n), since it is obtained by reversing the input sequence. Therefore, the system is causal.

The correct answer is y[n] = x[n] * x[n - 1]. This is a time-invariant system because the output y[n] depends only on the current and previous input values x[n] and x[n - 1], regardless of the specific time index n. The system does not change its behavior over time, making it time-invariant.

The given correct answer is "power signal". A power signal is a signal that has a finite power value, meaning that the average power over a given time interval is finite. This is in contrast to an energy signal, which has a finite energy value but may have infinite power. Since the question does not provide any additional information or context, it is reasonable to assume that δ(t), which represents the Dirac delta function, can be considered a power signal.

The given answer states that the sequence x[n] is non-periodic. This means that the sequence does not repeat itself after a certain number of terms. The reason for this conclusion is not provided in the question, so we cannot determine the exact explanation. However, it is possible that the sequence x[n] does not follow a specific pattern or does not have a recurring element, leading to its classification as non-periodic.