Question bank
1 Apply overlap-add and overlap save method to filter long data sequence 𝑥(𝑛) =
{3, 0, −2, 0, 2, 1, 0, −2, −1, 2,3} with the filter given by ℎ(𝑛) = {2, 1, 3}. Verify
your result using matrix method.
2 What is windowing technique in filter design? Design a HPF for the following
desired frequency response given by
−𝜋 𝜋
0 ≤𝑤≤
𝐻𝑑 (𝑒 𝑗𝑤 ) = { 3 3
𝜋
𝑒 −𝑗4𝑤 < |𝑤| ≤ 𝜋
3
Determine the filter coefficients ℎ(𝑛) if the window function is defined using
rectangular window.
3 What is twiddle factor? Write its two properties.
4 For a sequence x(n)= {1 3 5 7} Find its DFT X(K). And from the X(K) obtain its
IDFT x(n).
5 For the given H(z)
1
3 + 5𝑧 −1 + 3 𝑧 −2
𝐻(𝑧) =
7 5
2 + 3 𝑧 −1 − 3 𝑧 −2
Find the difference equation which represent the above system. Represent the
system by using DF-I and DF-II representation. Compare between DF1 and DF2
structure in terms of computational complexity.
6 Determine the Fourier Transform of the following signal
𝒕
𝐱(𝐭) = 𝐫𝐞𝐜𝐭(𝟔).
7 For two given sequences x1(n)={1,3,2,5} and x2(n)={3,2,4,2}, Find circular
convolution between x1(n) an x2(n). Verify circular convolution property of
DFT. Also find Linear convolution using circular convolution.
8 Discuss S-plane and z-Plane mapping.
9 Discuss about the relation between DTFT and Z-transform.
10 The system function of a causal LTI system is 𝐻(𝑧) =
1−𝑧 −1
. The input to this
3
1+ 𝑧 −1
4
1 𝑛
system is 𝑥(𝑛) = ( ) 𝑢(𝑛) + 𝑢(−𝑛 − 1). Find
3
(a) Impulse response of the system ℎ(𝑛)
(b) Find the output 𝑦(𝑛)
(c) Is the system stable
11 Suppose that we want to design a discrete time LTI system with the property
1 𝑛 1 1 𝑛−1
that if the input 𝑥(𝑛) = (2) 𝑢(𝑛) − 4 (2) 𝑢(𝑛 − 1) then the output y(𝑛) =
1 𝑛
(3) 𝑢(𝑛).
(a) Find the impulse response and frequency response of a discrete time LTI
system that has the foregoing property
(b) Find the difference equation relating 𝑥(𝑛) and 𝑦(𝑛) that characterizes
the system.
12 Design a LPF for the following desired frequency response given by
−𝜋 𝜋
𝑒 −𝑗4𝑤 ≤𝑤≤
𝐻𝑑 (𝑒 𝑗𝑤 ) = { 3 3
𝜋
0 < |𝑤| ≤ 𝜋
3
Determine the filter coefficients if the window function is defined using
rectangular window.
13 Find the exponential Fourier series representation of a sawtooth wave having a
period of 4 units and maximum amplitude of 3 units (whose one period is given
below)
14 Check whether the system 𝑦(𝑛) = cos[𝑥(𝑛)] is
(a) static/dynamic
(b) Linear/Nonlinear
© Causal/ Non-causal
d. Time variant/ Time invariant
e. Stable/ unstable
15 Find the Fourier transform of the following signals
(a) 𝑥(𝑡) = 𝑠𝑔𝑛(𝑡)
(b) X(t)=u(t)
16 Test the stability of the LTI systems whose impulse response is
ℎ(𝑛) = 0.3𝑛 𝑢(𝑛) + 0.4𝑛 𝑢(𝑛)
17 For an LTI system the impulse response is given as ℎ(𝑛) = 3𝛿(𝑛 + 1) −
2𝛿(𝑛 − 1) + 2𝛿(𝑛 − 2) . Estimate the response of the system for the input
𝑥(𝑛) = 3𝛿(𝑛 + 1) + 2𝛿(𝑛) + 2𝛿(𝑛 − 1) using concentric circle method.
18 1
Consider a Causal LTI system with frequency response 𝐻(𝑤) = 𝑗𝑤+4 . For a
particular input 𝑥(𝑡) the system is observed to produce the output 𝑦(𝑡) =
𝑒 −3𝑡 𝑢(𝑡) − 𝑒 −4𝑡 𝑢(𝑡). Determine the input to the system.
