STUPIDSID
1(a)
Define : Signal. Find the fundamental periods (T for continuous-time signals, N for discrete- time signals) of the following periodic signals.
1. x(t)=cos(13πt)+2sin(4πt)
2. x[n]ej7.351πn
1. x(t)=cos(13πt)+2sin(4πt)
2. x[n]ej7.351πn
7 M
1(b)
Define: System.
Determine whether the system y(t) = t x(t) is
1. Memoryless
2. Linear
3. Time invariant
4. Causal
5. BIBO stable. Justify your answers.
Determine whether the system y(t) = t x(t) is
1. Memoryless
2. Linear
3. Time invariant
4. Causal
5. BIBO stable. Justify your answers.
7 M
2(a)
Compute the convolution y(n) = x(n) * h(n)
1. x[n] =δ[n] - δ[n-2], h[n] = u[n]
2. x[n] =u[n], h[n] = u[n]
1. x[n] =δ[n] - δ[n-2], h[n] = u[n]
2. x[n] =u[n], h[n] = u[n]
7 M
Solved any one question from Q.2(b) & Q.2(c)
2(b)
Determine the trigonometric Fourier series for signal given below \(\displaystyle\delta _{T_0}(t)=\sum^{\infty}_{k=-\infty}\delta (t-kT_0)\)
7 M
2(c)
Determine the complex exponential Fourier series for
1.cos(ωt)
2.sin2t
1.cos(ωt)
2.sin2t
7 M
Solved any one question from Q.3 & Q.4
3(a)
Define: The continuous time Fourier transform.
State and prove Time shifting and Duality properties of continuous time Fourier transform.
State and prove Time shifting and Duality properties of continuous time Fourier transform.
7 M
3(b)
Find the Z transform of
1. δ(n)
2.u[n]
3. nanu[n]. (1+2+4 Marks)
1. δ(n)
2.u[n]
3. nanu[n]. (1+2+4 Marks)
7 M
4(a)
Define: The Z transform.
State and prove Time shifting and Time reversal properties of Z transform.
State and prove Time shifting and Time reversal properties of Z transform.
7 M
4(b)
Find the continuous time Fourier transform
1. δ(t)
2. e-atu[t], a>0
3. u[t]. (1+2+4 marks)
1. δ(t)
2. e-atu[t], a>0
3. u[t]. (1+2+4 marks)
7 M
Solved any one question from Q.5 & Q.6
5(a)
Using power series expansion technique find the inverse Z transform of
\[X(z)=\frac{1}{1-az^{-1}},\left | z \right |>\left | a \right |\].
\[X(z)=\frac{1}{1-az^{-1}},\left | z \right |>\left | a \right |\].
7 M
5(b)
The output y [n] of a discrete-time LTI system is found to be \[2\left ( \frac{1}{3} \right )^n\] u[n] when the input x[n] is u[n] .Find the impulse response h[n] of the system.
7 M
6(a)
Using the partial fraction expansion technique find the inverse Z transform of
\[X(Z)=\dfrac{z}{2z^2-3z+1},\left | z \right |<\dfrac{1}{2}\[ .
\[X(Z)=\dfrac{z}{2z^2-3z+1},\left | z \right |<\dfrac{1}{2}\[ .
7 M
7(a)
Define: Convolution Sum.
Show that
1. x[n]*&delta[n]=x[n]
2. x[n]*&delta[n-n0]=x[n-n0
3. x[n]*u[n-n0]=\[\sum ^{n-n_0}_{k=-\infty }\] x [k]
Show that
1. x[n]*&delta[n]=x[n]
2. x[n]*&delta[n-n0]=x[n-n0
3. x[n]*u[n-n0]=\[\sum ^{n-n_0}_{k=-\infty }\] x [k]
7 M
7(b)
A Continuous-time periodic signal x(t) is real valued and has a fundamental period T = 8. The non-zero Fourier series coefficients for x(t) are
a1 = a-1 = 2, a3= a*-3=4j.
Express x(t) in the form
\[x(t)=\sum ^\infty _{k=0}A_k\cos(\omega _kt+\phi_k)\].
a1 = a-1 = 2, a3= a*-3=4j.
Express x(t) in the form
\[x(t)=\sum ^\infty _{k=0}A_k\cos(\omega _kt+\phi_k)\].
7 M
8(a)
Define the condition for LTI system to be stable. Which of the following
impulse responses correspond to stable LTI systems.
1. h1 (t) = e-(1-2j)t u(t)
2. h2(n)=3nu[-n+10]
1. h1 (t) = e-(1-2j)t u(t)
2. h2(n)=3nu[-n+10]
7 M
8(b)
Define Laplace transform. Prove linearity property for Laplace transform. State how ROC of Laplace transform is useful in defining stability of systems.
7 M
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