Unit 2 - Practice Quiz

LTI stands for Linear Time-Invariant, which are two fundamental properties that describe a particular class of systems. Linearity refers to the superposition principle, and time-invariance means the system's behavior does not change over time.

A system is linear if it obeys the superposition principle, which combines the properties of additivity (response to a sum of inputs is the sum of responses) and homogeneity (scaling the input scales the output by the same factor).

The convolution sum is the fundamental operation for finding the output of a discrete-time LTI system. It involves summing the product of the input signal and a time-reversed, shifted version of the impulse response.

By definition, the impulse response of an LTI system is the output produced when the input is the Dirac delta function . This response completely characterizes the system.

Causality is a property of real-world systems where the effect cannot precede the cause. Therefore, the output at a specific time can only depend on the input for times .

The commutative property of convolution states that the order of the two signals being convolved does not affect the result. We can interchange the roles of the input and the impulse response.

Continuous-time signals are functions of a continuous variable, conventionally denoted by 't' for time. Thus, is the standard notation. is used for discrete-time signals.

For a causal system, the output cannot depend on future inputs. This means the impulse response (the response to an impulse at t=0) must be zero for all negative time, i.e., for .

For a time-invariant system, the characteristics do not change over time. If an input produces an output , then a shifted input will produce an identically shifted output .

The convolution integral defines the output of a continuous-time LTI system. It involves integrating the product of one signal and a time-reversed, shifted version of the other signal.

The associative property allows the grouping of convolution operations to be changed. It is particularly useful for analyzing systems connected in cascade, where the overall impulse response is the convolution of individual impulse responses.

The output at time depends on the current input and a past input . Since it does not depend on future inputs (e.g., ), the system is causal.

The convolution operation is fundamental to LTI systems. Software tools provide built-in functions, commonly named conv() (in MATLAB) or convolve() (in SciPy), to perform this calculation.

BIBO stands for Bounded-Input, Bounded-Output. It is a key definition of stability. If any input signal that stays within a finite range () results in an output that also stays within a finite range (), the system is BIBO stable.

The impulse response is the output when the input is an impulse, . Substituting this into the equation gives . Therefore, .

In a cascade connection, the output of the first system becomes the input to the second. The overall relationship between the original input and final output is found by convolving their individual impulse responses.

Difference equations relate the current output of a discrete-time system to past outputs and current/past inputs. They are the discrete-time counterpart to differential equations.

This system is time-varying. A delay in the input produces an output , which is not the same as a delayed output . The system's behavior changes with time.

Correlation is a mathematical operation that measures the similarity of two signals as a function of the time lag applied to one of them. It is widely used in applications like pattern recognition and signal detection.

A memoryless (or static) system's output depends solely on the input value at that specific time. An example is a simple resistor where .

The output is the convolution of and . Using the convolution sum formula, . For , we have . The non-zero values of are at and . So, . The impulse response is for . Thus, and . The input is . Therefore, .

The output is the step response of the system, which can be found by integrating the impulse response or by convolution. for . Evaluating the integral gives: .

Causality requires for . The term is non-zero for , so the system is non-causal. For BIBO stability, the impulse response must be absolutely summable. The sum of the first part is , which converges. The sum of the second part is , which also converges. Since both parts converge, the system is stable.

We can solve this recursively. Assume . For the input , for .
.
.
.
.

Linearity: If the input is , the output is . The system is linear.
Time-Invariance: Let the input be delayed by , so . The output is . Now, delay the original output by : . Since , the system is time-variant.

For systems in cascade, the overall impulse response is the convolution of the individual impulse responses: . The convolution is given by . For , this integral becomes . For , the integral is 0. Therefore, the result is , which is the unit ramp function .

The duration of a signal is the length of the time interval over which it is non-zero. The duration of the rectangular pulse is . The duration of the rectangular pulse is . The duration of the convolution of two finite-duration signals is the sum of their individual durations. Therefore, the duration of is .

The length of the convolution of two finite-duration sequences of length and is given by the formula . In this case, and , so the length of is .

To find the impulse response , we solve the equation with . The equation becomes . For a first-order system of the form , the impulse response for a causal system is . Here, , so the impulse response is .

The cross-correlation is defined as . The convolution is defined as . If we let , then . The convolution becomes . Therefore, cross-correlation is equivalent to convolving one signal with the time-reversed version of the other.

The output of this system is . This operation, which computes the difference between consecutive input samples, is known as a first-difference system, the discrete-time equivalent of a differentiator.

This system is an accumulator.
Linear: The summation operator is linear.
Time-Invariant: A shift in the input results in a corresponding shift in the output sum, so it is time-invariant.
Causal: The output at time depends only on the present and past inputs (from to ), so it is causal.
Has Memory: The output at time depends on past values of , so it is not memoryless.

The discrete-time signal is obtained by substituting .
.
This simplifies to . The Nyquist rate for this signal is Hz = 20 Hz. Since the sampling rate (15 Hz) is less than the Nyquist rate, the second component is aliased, but the mathematical substitution remains the same. Note that , so aliasing occurs, but the direct substitution gives this answer.

The impulse response is the derivative of the step response .
.
Using the product rule for differentiation: .
.
Using the sifting property, .
So, .

The convolution sum is for .
.
This is a finite geometric series with sum . This can be written as . The result is valid for , so we multiply by .

The condition for Bounded-Input, Bounded-Output (BIBO) stability for a continuous-time LTI system is that its impulse response must be absolutely integrable. This means that the integral of the absolute value of over all time must be a finite value. The other options are not necessary conditions for stability; for example, corresponds to a stable system but is not of finite duration.

The impulse response is found by setting , which gives . This impulse response is non-zero only at and . Since the impulse response has a finite number of non-zero terms, the system is a Finite Impulse Response (FIR) system. It is also a causal, linear system.

The auto-correlation function is defined as . Let's evaluate . By a change of variable , we get and . The integral becomes , which is identical to the definition of . Thus, the auto-correlation function always has even symmetry.

Let's test .
Linearity: . It is linear.
Time-Invariance: A delayed input yields an output . A delayed output is . Since these are not equal, the system is time-variant.
is non-linear. is LTI. is non-linear (fails homogeneity).

This question describes the two defining properties of an LTI system. The fact that scaling the input by scales the output by is part of the linearity property (specifically, homogeneity). The fact that shifting the input by shifts the output by is the definition of time-invariance. A system that possesses both properties is a Linear Time-Invariant (LTI) system.

The convolution sum is . Since for and is zero otherwise, the sum becomes . For the region , the term is always 1 for any in the summation range (). The sum is . This is a finite geometric series with terms, first term and common ratio . The sum is given by .

The convolution integral is . The term is 1 for . The term is 1 for , which means . For the integrand to be non-zero, both conditions must be met: and . This means the upper limit of integration is . The integral is . For the integral to be non-zero, the upper limit must be greater than . If (i.e., ), the upper limit is . The integral is . If (i.e., ), the upper limit is $0$. The integral is . This is a saturated constant value. Therefore, the output is non-zero and not saturated (i.e., still changing with t) for .

Causality requires for . The term is 1 for , i.e., . This means is non-zero for (e.g., ). Thus, the system is non-causal.

BIBO stability requires the impulse response to be absolutely integrable, . The integral is . This evaluates to . Since the integral converges to a finite value, the system is stable.

The easiest way to solve this is using the Laplace Transform. The transfer function is . This indicates a pole-zero cancellation. The impulse response is the inverse Laplace transform of , which is . The zero-state response is the convolution of the input with the impulse response: .
for .
.

1. Linearity: The integral operation is linear. Scaling or summing inputs results in a scaled or summed output. The system is linear.
2. Causality: The output at time is determined by the integral of the input up to . It does not depend on future values of the input. The system is causal.
3. Time-Invariance: Let's check the response to a shifted input . The output is . The shifted original output is . Let in the expression for . Then . Comparing with , the factor in is different from the factor in . Thus, , and the system is time-variant.

A key property of LTI systems is that the sum of the output sequence is the product of the sums of the input and impulse response sequences: .
First, calculate the sum of the input signal: . This is a geometric series which sums to .
We are given . Using the property, .
This implies . Since the system is causal, for , so .
The sum can be split as .
We are given . So, .
Therefore, .

For BIBO stability of an LTI system, the necessary and sufficient condition is that the impulse response is absolutely integrable: . For this system, we need to evaluate . This is a well-known improper integral that diverges. To see this, consider the integral over intervals for integer . The area under each lobe of decreases like , and the sum of these areas behaves like the harmonic series, which diverges. The condition as is necessary but not sufficient for stability. The convergence of (which does converge to 1/2, this is the Dirichlet integral) is also not sufficient; the integral of the absolute value must converge. The system is causal due to the term. Therefore, the system is unstable.

To find the impulse response , we solve the equation for with zero initial conditions ( for ). The characteristic equation is , which is . This indicates a repeated root at . The general form of the homogeneous solution is . We find the coefficients by calculating the first few values of from the difference equation.
.
For : .
For : .
Now, we use these initial values with the general solution form:
For : . So, .
For : . So, .
Therefore, the impulse response is .

It's easier to first find the step response . for . Using linearity and time-invariance, the response to is .
.
We analyze in three regions:

1. For , .
2. For , . The derivative is , so the function is increasing in this interval.
3. For , . The derivative is , so the function is decreasing.
   Since the function increases until and then decreases, the maximum value occurs at . The value is . This is the same as the correct option when factored.

1. Linearity: The system is a weighted sum of the input values. Let and be the outputs for inputs and . The output for is . The system is linear.
2. Time-Invariance: Let's test for a shift. Let the input be . The output is . Now consider the shifted original output, . Clearly, due to the term vs . The system is time-variant. The non-linear indexing of the input signal makes the system time-variant.

First, find the inverse system. Using Laplace transforms, . The inverse system's transfer function is . To find the impulse response , we take the inverse Laplace transform: .
Now, let's analyze the properties of .
Causality: The impulse response consists of an impulse and its derivative, both at . It is zero for . Thus, the inverse system is causal.
Stability: For BIBO stability, we must have . The integral of is 1, but the integral of is unbounded. An input like a step function to a differentiator produces an impulse, which is bounded, but an input like would produce an unbounded output. A system containing a pure differentiator is unstable. Therefore, the inverse system is causal but unstable.

In the overlap-save method, we use an FFT of size . The impulse response has length . Each block of input data has length . The first points of each output block are discarded.
Given: (input length), (filter length), (FFT size).
First, we compute the FFT of the zero-padded impulse response once: . This is 1 FFT.
The number of valid output points per block is .
The total length of the output signal is .
The number of blocks, , required to produce at least points is blocks.
For each of these 7 blocks of input, we perform one FFT and one IFFT.
Total FFTs/IFFTs = 1 (for ) + 7 (for input blocks) + 7 (for IFFTs) = 15.

The transfer function of the given system is . For an all-pass filter, the poles and zeros must be conjugate reciprocals of each other. The general form for a first-order all-pass filter is . The magnitude of this is . Let's rewrite the options in terms of .
The general equation is , so .

First, find the impulse response of the first system (the echo model). The output is the convolution of the input with the impulse response . From the equation , we can identify .
The second system's impulse response is given as .
For a cascaded system, the overall impulse response is the convolution of the individual impulse responses: .
.
Using the distributive property of convolution:

Using the sifting property ():

.
This represents a system that passes the original signal and subtracts a scaled version of the signal delayed by .

First, let's find the output for the given input .
.
As , , so . The output is clearly bounded for this specific bounded input.

This is a deconvolution problem. We can use polynomial multiplication in the Z-domain. , so .


We need to perform polynomial long division:

Dividing $1$ by gives $1$. .
Subtract: . Bring down .
We have . Divide by $1$ gives . .
Subtract: . Bring down .
We have . Divide by $1$ gives . .
Subtract: .
We have . Divide by $1$ gives . .

.



New Question: Let where and . Find .
. . . . So . . This is a good hard question. Let's create options for this. Options: A) , B) , C) , D) .

The overall impulse response is the N-fold convolution of with itself: (N times). It is easier to work in the Z-domain. The transfer function of one system is . For N cascaded systems, the total transfer function is . We need to find the inverse Z-transform of this expression. We can use the transform pair: . In our case, and . So the inverse Z-transform is . The binomial coefficient accounts for the repeated convolutions.