Description
I Paper Assignment
- Explain whether the following systems are linear, time-invariant, causal, and stable.
- y[n] = x[-n]
- y[n] = cos(πn)𝑥[𝑛]
- A linear system L generates output signals: y1[n], y2[n] in response to the input signals x1[n], x2[n] respectively. Explain whether the system L is time invariant.
x1[n] y1[n]
x2[n] y2[n]
- Given the z-transform pair x[n] ↔ X(z) = with ROC: |z| < 2, use the z-transform properties to determine the z-transform of the following sequences:
- y[n] = (1)n x[n]
3
- y[n] = x[n]*x[-n] ( * denotes convolution)
- y[n] = nx[n]
- A causal LTI system has impulse response h[n], for which the z-transform is
1 + 𝑧−1
𝐻(𝑧) = (1 − 0.5𝑧−1)(1 + 0.25𝑧−1)
- Draw the pole-zero plot of H(z) and specify its ROC.
- Explain whether the system is stable?
- Find the impulse response h[n] of the system.
- Use the method of partial fraction expansion to determine the sequences corresponding to the following z-transforms:
- X(z) = 𝑧3+ 2𝑧2𝑧+ 5𝑧+ 1 , |z| >1.
4 4
- X(z) = (𝑧2−𝑧 1)2 , |z| <0.5.
- A function called autocorrelation for a real-valued, absolutely summable sequence x[n], is defined as
𝑟𝑥𝑥[ℓ] ≜ ∑𝑛 𝑥[𝑛]𝑥[𝑛 − ℓ].
Let X(z) be the z-transform of x[n] with ROC α < |z| < β.
- Show that the z-transform of rxx[ℓ] is given by Rxx(z) = X(z)X(z−1).
- Let x[n] = anu[n], |a| < 1. Determine Rxx(z) and sketch its pole-zero plot and the
ROC.
- Determine the DTFT of following signals:
1 𝑛 𝜋𝑛
- x1[n] = ( ) 𝑐𝑜𝑠 ( ) 𝑢[𝑛 − 2]
4 4
- x3[n] = 𝑠𝑖𝑛(0.1𝜋𝑛)(𝑢[𝑛] − 𝑢[𝑛 − 10])
- Let x[n] and y[n] denote complex sequences and X(ejω) and Y(ejω) their respective
Fourier transforms
- Determine, in terms of x[n] and y[n], the sequence whose Fourier transform is
X(ejω)Y*(ejω).
- Using the result in part (a), show that
. (eq.7b)
(eq.7b is a more general form of Parseval’s theorem, as mentioned in lecture slide ch4 p17.)
- Using (eq.7b), determine the value of the sum
𝜋𝑛/3)
𝜋𝑛
(Hint: Check what is the expression of an inverse Fourier transform of a rectangular pulse mentioned in lecture slide ch4 p23.)
