PARISUTHAM INSTITUTE OF TECHNOLOGY AND SCIENCE
Department of ECE
AY 2015-16
IV Year-CSE /VII Sem.
CAT 1
CS2403 DIGITAL SIGNAL PROCESSING
Duration: 3 Hours
Max. Marks: 100
Answer all the questions
PART-A (10x2 marks =20 marks)
1. State the sampling theorem.
 A continuous time signal can be completely represented in its samples and recovered
back if the sampling frequency 𝑓𝑠 ≥ 2𝜔.
 Here 𝑓𝑠 is the sampling frequency and 𝜔 is the maximum frequency.
2. State any properties of LTI systems.
Linearity =>
𝑇[𝑎1 𝑥1 (𝑛) + 𝑎2 𝑥2 (𝑛)] = 𝑎1 𝑇[𝑥1 (𝑛)] + 𝑎2 𝑇[𝑥2 (𝑛)]
Time Shifting =>
𝑌(𝑛) = 𝑇[𝑥(𝑛)], then
𝑌(𝑛 − 𝐾) = 𝑇[𝑥(𝑛 − 𝐾)]
= 𝑍 −𝐾 𝑇[𝑥(𝑁)]
𝑍 −𝐾 = 𝑑𝑒𝑙𝑎𝑦
3. Determine the Z transform for 𝒙(𝒏) = −𝒏𝒂𝒏 𝒖(−𝒏 − 𝟏).
The Z-Transform for given values is
𝑥(𝑛) = −𝑛𝑎𝑛 𝑢(−𝑛 − 1)
𝑎𝑧 −1
𝑥(𝑧) =
(1 − 𝑎𝑧 −1 )2
𝑎𝑧
𝑥(𝑧) =
(∴ |𝑧| < |𝑎|)
(𝑧 − 𝑎)2
4. Find whether the signal 𝒚 = 𝒏𝟐 𝒙(𝒏) is linear.
𝑦(𝑛) = 𝑛2 𝑥(𝑛)
we have 𝑌1 (𝑛) = 𝑇[𝑥1 (𝑛)] = 𝑛2 𝑥1 (𝑛)
𝑌2 (𝑛) = 𝑇[𝑥2 (𝑛)] = 𝑛2 𝑥2 (𝑛)
The weighted sum of output is
𝑎1 𝑇[𝑥1 (𝑛)] + 𝑎2 𝑇[𝑥2 (𝑛)] = 𝑎1 𝑛2 𝑥1 (𝑛) + 𝑎2 𝑛2 𝑥2 (𝑛)
The output to the weighted sum of input is
𝑌3 (𝑛) = 𝑇[𝑎1 𝑥1 (𝑛) + 𝑎2 𝑥2 (𝑛)] = 𝑛2 𝑎1 𝑥1 (𝑛) + 𝑛2 𝑎2 𝑥2 (𝑛)
Hence the system is NON-LINEAR
5. State and prove parseval’s theorem.
If DFT [x(n)] = X(k) and
DFT [y(n)] = Y(k) then
𝑁−1
𝑁−1
∑ 𝑥(𝑛) 𝑦
∗ (𝑛)
𝑛=0
𝑛=0
𝑁−1
∑ 𝑥(𝑛) 𝑦
𝑛=0
1
= ∑ 𝑋(𝑘)𝑌 ∗ (𝑘)
𝑁
∗ (𝑛)
𝑁−1
𝑁−1
𝑛=0
𝑛=0
1
1
= ∑ 𝑥(𝑛) [ ∑ 𝑌 ∗ (𝑘)𝑒 −𝑗2𝜋𝑘𝑛⁄𝑁 ]
𝑁
𝑁
𝑁−1
𝑁−1
𝑛=0
𝑛=0
1
1
= ∑ 𝑌 ∗ (𝑘) [ ∑ 𝑥(𝑛)𝑒 −𝑗2𝜋𝑘𝑛⁄𝑁 ]
𝑁
𝑁
𝑁−1
1
= ∑ 𝑌 ∗ (𝑘) 𝑋(𝑘)
𝑁
𝑛=0
Hence Proved
6. Compute the DFT of the four point sequence 𝒙(𝒏) = {𝟎, 𝟏, 𝟐, 𝟑 … }.
𝑁−1
𝑋(𝑘) = ∑ 𝑥(𝑛)𝑒 −𝑗2𝜋𝑘𝑛⁄𝑁
𝑛=0
𝑋(𝑘) = {6, −2 + 2𝑗, −2, −2 − 2𝑗}
7. Draw the basic butterfly of the Radix-4 DIT algorithm.
8. What is phase factor or twiddle factor?
The complex number 𝑊𝑁 is called phase factor or twiddle factor.
𝑊𝑁 = 𝑒
−𝑗2𝜋
𝑁
𝑊0 = 1
𝑊1 = 0.707 − 𝑗0.707
𝑊2 = −𝑗
𝑊3 = −0.707 − 𝑗707
9. Find the DFT for 𝒙(𝒏) = {𝟏, −𝟏, 𝟏, −𝟏}.
The 4-point DFT of x(n) is given by
3
𝑋(𝐾) = ∑ 𝑥(𝑛) 𝑒 −𝑗2𝜋𝑛𝑘⁄4
𝑛=0
= 𝑥(0) + 𝑥(1)𝑒 −𝑗𝜋𝑘⁄2 + 𝑥(2)𝑒 −𝑗𝜋𝑘 + 𝑥(3)𝑒 −𝑗3𝜋𝑘⁄2
∴ 𝑥(0) = 1, 𝑥(1) = −1, 𝑥(2) = 1, 𝑥(3) = −1
= 1 − 𝑒 −𝑗𝜋𝑘⁄2 + 𝑒 −𝑗𝜋𝑘 − 𝑒 −𝑗3𝜋𝑘⁄2 ; 𝑓𝑜𝑟 𝑘 = 0, 1, 2, 3.
10. What are the requirements for converting a stable analog filter into a stable digital filter?

Mapping of desired digital filter specifications into equivalent analog filter.

Analog transfer function is derived for the analog filter.

Then, analog prototype filter is transformed into equivalent digital filter transfer function.
PART –B (5x 16 marks = 80 marks)
11. (a) (i) Determine the Z-transform and compute the ROC of the following signal
𝟏 𝒏
𝟑
𝟏 𝒏
𝟑
𝒙(𝒏) = ( ) 𝒖(−𝒏) + ( ) 𝒖(𝒏).
(16)
∞
𝑋(𝑧) = ∑ 𝑥(𝑛)𝑍 −𝑛
𝑛=−∞
∞
𝑋(𝑧) = ∑ 𝑥(𝑛)𝑍
𝑛=−∞
∞
−𝑛
𝑋(𝑧) = [− ∑(1⁄3)
𝑛=0
−𝑛
∞
+ ∑ 𝑥(𝑛)𝑍 −𝑛
𝑛=−∞
∞
𝑛
𝑍 +𝑛 ] + [− ∑(1⁄3) 𝑍 −𝑛 ]
𝑛=0
1
𝑍
+
(1
3𝑧 − 1 𝑍 − ⁄3)
(OR)
(b) (i) Compute the linear convolution of the signals x(n)={1,2,3,4,5,3,-1,-2}&
h(n)={3,2,1,4} using Tabulation method & Matrix method
(8)
 Tabulation method: y(n)={3,8,14,24,34,35,24,15,7,-6,-8} (4 marks)
 Matrix Method: y(n)={3,8,14,24,34,35,24,15,7,-6,-8} (4 marks)
(ii) Find the output of the linear convolution using Circular Convolution
x(n)={1,-3,5,-7,9,-11} & h(n)={-4,8,-16} [ Matrix Method]
.
(8)
y(n)={-4,20,-60,116,-172,228,-232,176}
ANS:
12. (a) (i) State and explain sampling theorems.
(8)
Definition – 2marks
Explanation – 2marks
𝑥(𝑡) = 𝑠𝑖𝑛 𝑢𝑡
𝑥(𝑛𝑇) = 𝑠𝑖𝑛 Ω𝑛𝑇 = 𝑠𝑖𝑛 𝜔𝑛 → 2𝑚𝑎𝑟𝑘𝑠
𝑓𝑠 ≥ 2𝑓𝑚 → 2𝑚𝑎𝑟𝑘𝑠
(ii) Find the Z-transform auto correlation function.
Definition – 2marks
(8)
∞
𝛾𝑥𝑥 (𝑙) = ∑ 𝑥(𝑛)𝑥(𝑛 − 𝑙) − > 2𝑚𝑎𝑟𝑘𝑠
𝑛=−∞
∞
∞
𝑌(𝑍) = ∑ [ ∑ 𝑥(𝑘)𝑥(𝑛 − 𝑘)] 𝑧 −𝑛 → 2𝑚𝑎𝑟𝑘𝑠
𝑛=−∞ 𝑘=−∞
𝛾𝑥𝑥 (𝑙) = 𝑋(𝑍)𝑋(𝑍 −1 ) → 2𝑚𝑎𝑟𝑘𝑠
(OR)
(b) (i) Suppose a LTI system with input x(n) & output y(n) is characterized by its
unit sample response h(n)= (0.8)n u(n). Find the response y(n) of the system having
the input x(n)= u(n).
(10)
ANSWER:
h(n)= (0.8)n u(n) => an u(n)=
Y(z)=H(z).X(z)=
𝒛
.
𝑧
𝒛−𝟎.𝟖 𝑧−1
𝒛
; H(z)=
𝒛−𝒂
𝒛
𝒛−𝟎.𝟖
; X(z)=
𝑧
𝑧−1
(taking inverse Z-Transform using partial fraction method)
Y(n)= (-4. (0.8)n +5) u(n)
(ii) A causal system represented by the difference equation,
𝟏
𝒚(𝒏) + 𝟒 𝒚(𝒏 − 𝟏) = 𝒙(𝒏) +
𝟏
𝟐
𝒙(𝒏 − 𝟏). Compute its transfer function H(z).
ANSWER: Taking inverse transform and compute the H(z)= Y(z)/ X(z)
𝟏
H(z)=
(6)
𝟏+(𝟐)𝒛^−𝟏
𝟏
𝟏+(𝟒)𝒛^−𝟏
13. (a) Given 𝒙(𝒏) = {𝟎, 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕} find 𝑿(𝒌) using DIT FFT algorithm.
(16)
X(K)= {28, -4+j 9.656, -4+j4, -4+j 1.656, -4 ,-4-j 1.656, -4-j 4, -4-j 9.656}
(OR)
(b) Given 𝒙(𝒏) = 𝟐𝒏 compute 𝑿(𝒌) using DIF FFT algorithm for N = 8 and 𝒏 ≥ 𝟎
(16)
14. (a) (i) Derive the key equations of radix-2 DIF FFT algorithm and draw the relevant flow
graph taking the computation of an 8 point DFT for your illustration.
(8)
 Explanation – 2marks
 Graph diagram – 6marks
(ii)Compute the FFT for the sequence 𝒙(𝒏) = 𝒏 + 𝟏 where N = 8 using the inplace radix 2
decimation frequency algorithm.
(8)
(OR)
(b) (i) List out all the properties of DFT.
Definition – 2marks
(8)
Radix 2 flow graph – 2marks
Properties and explanation – 4marks
(ii) Determine the circular convolution of the sequences
(8)
𝒙𝟏 (𝒏) = {𝟏
⏟ , 𝟏, 𝟐, 𝟏}
↑
𝒙𝟐 (𝒏) = {𝟏
⏟ , 𝟐, 𝟑, 𝟒}
↑
1
[1
2
1
1
1
1
2
2
1
1
1
𝑦(0)
1 1
2 ] [2] = [𝑦(1)]
1 3
𝑦(2)
1 4
𝑦(3)
𝒚(𝒏) = {𝟏𝟑, 𝟏𝟒, 𝟏𝟏, 𝟏𝟐}
15. (a) Determine the cascade and parallel realization for the system, described by the system
function
𝑯(𝒛) =
(16)
𝟏𝟎(𝟏 − (𝟏⁄𝟐)𝒛−𝟏 )(𝟏 − (𝟐⁄𝟑)𝒛−𝟏 )(𝟏 + 𝟐𝒛−𝟏 )
[(𝟏 − (𝟑⁄𝟒)𝒛−𝟏 )(𝟏 − (𝟏⁄𝟖)𝒛−𝟏 )(𝟏 − (𝟏⁄𝟐 + 𝒋 𝟏⁄𝟐)𝒛−𝟏 )(𝟏 − (𝟏⁄𝟐 − 𝒋 𝟏⁄𝟐)𝒛−𝟏 )]
𝑪𝒂𝒔𝒄𝒂𝒅𝒆:
Parallel:
(OR)
(b) Determine H(z) for a butterworth filter satisfying the following constraints
𝝅
√𝟎. 𝟓 ≤ |𝑯(𝒆𝒋𝝎 )| ≤ 𝟏
𝟎≤𝝎≤
𝟐
𝒋𝝎
𝟑𝝅
⁄𝟒 ≤ 𝝎 ≤ 𝝅
|𝑯(𝒆 )| ≤ 𝟎. 𝟐
with T = 1s. Apply impulse invariant transformation.
(16)
Ω𝑝 = 𝜔𝑝 ; Ω𝑠 = 𝜔𝑠 → 2𝑚𝑎𝑟𝑘𝑠
𝜆 = 4.989; 𝜖 = 1 → 2𝑚𝑎𝑟𝑘𝑠
𝑁≥
𝜆
log 𝜖
Ω
log Ω 𝑠
= 4 → .4𝑚𝑎𝑟𝑘𝑠
𝑝
𝐻(𝑆) =
𝐻(𝑧) =
1
→ 2𝑚𝑎𝑟𝑘𝑠
(𝑠 2 + 0.76537𝑠 + 1)(𝑠 2 + 1.8477𝑠 + 1)
1.454 + 0.1839𝑧 −1
−1.454 + 0.2307𝑧 −1
+
→ 6𝑚𝑎𝑟𝑘𝑠
1 − 0,387𝑧 −1 + 0.055𝑧 −2 1 − 0.1322𝑧 −1 + 0.301𝑧 −2
Faculty Signature:
Faculty Name: S.SHANMUGA PRIYA
HOD Signature