Signals and Systems — Summer 2023

Define a signal and a system stating examples of each.

Find the following sequences are periodic or not. If periodic, find the fundamental period.

x(t) = sin15𝜋t +sin20𝜋t ii) x(n) = ej(𝜋/4)n

Explain the Classification of Systems. State Sampling Theorem. Determine the sampling rate of a television

03 signal having a bandwidth of 4.5 MHz.

Compare Energy and Power Signals with example of each.

For a given systems, determine and justify whether each of them is: i) Memoryless ii) Stable iii) Causal iv) Linear and v) TIV

y(n) = ex(n) ii) y(t) = x(t2)

For x(n) = {3,2,1,0,1,2,3} and y(n) = {−1,−1,−1,−1,0,1,1,1,1} Compute: i) x(n−2)+y(n+2) ii) x(n+2) y(6−n) iii) x(3−n) y(n)

Define Laplace Transform and determine the Laplace Transform of x(t) = t2e−3tu(t)

State and prove time scaling property of Laplace Transform.

Compute Linear Convolution using graphical method for x(n) = ℎ(n) = {1,2,−1,3}

Define Fourier Transform and give its relation with Laplace Transform.

State and prove differentiation property of Fourier Transform.

For an LTI system with the unit impulse response ℎ(t) = e−2tu(t), determine the output to the input x(t) = e−tu(t)

Define DTFT and give its relation with DFT.

State and prove the time and frequency shifting properties of Fourier Transform.1

Compute Circular Convolution of the sequence x(n) = ℎ(n) = {1,2,−1,3} and compare its result with linear convolution.

Find the Fourier Transform of x(t) = t2e−3tu(t)

Compute the DFT of x(n) = {−2,−1,0,1}

List the properties of ROC of Ztransform and find the inverse Z- Transform of X(Z) = log (1+aZ−1)

Define Z Transform and its ROC.

State and prove differentiation property of Z Transform and find the Z transform of x(n) = n u(n)

Find the inverse Z-Transform of

Derive the relationship between Ztransform and Fourier transform.

State and prove time reversal property of Ztransform and determine the Ztransform of x(n) = anu(−n−1)

Find the inverse Z- Transform of