Digital Signal Processing

(BEC502)
(2022 scheme)

Module-2

Module 3
 Channel Name: Engg-Course-Made-Easy SUBSCRIBE

Subject: Signals and Digital Signal Processing

BEC502
Click on corresponding You-tube video link to for the topic.

MODULE 2:Discrete Fourier Transform and DFT Properties

What is DFT. Derive the expression for DFT. How to reconstruct signal.
1
(Frequency Domain sampling and Reconstruction of Discrete Time Signals) https://youtu.be/4g4-Sst97VA
Compute 4-point DFT of the sequence given by
2 https://youtu.be/WhffJkOrR3o
Compute 16-point DFT of the sequence
3 https://youtu.be/8pSmLpJJjRI
x(n)=8; 0 ≤ n ≤ 15
Compute N point DFT of the sequence
4 2𝜋𝑘0 𝑛 https://youtu.be/f38QqRMZAvk
𝑥(𝑛) = 𝑠𝑖𝑛 ( ), 0 ≤ n ≤ N-1
𝑁

Compute N point DFT of the following signal

5
https://youtu.be/KVQvRPgDUlE

Compute N point DFT of the sequence https://youtu.be/nNHvE0zknkc

6 2𝜋𝑘0 𝑛
𝑥(𝑛) = 𝑐𝑜𝑠 ( ), 0 ≤ n ≤ N-1
𝑁
 Matrix method of computing DFT (DFT as a Linear Transformation)
OR
7 Illustrate how the DFT & IDFT can be viewed as a Linear Transformations on sequences {𝑥(𝑛)} and {𝑋(𝑘)} https://youtu.be/ciBKcVDqmPE
respectively.

8 Compute 4 – point DFT of the signal 𝑥(𝑛)={0,1,2,3} using DFT matrix https://youtu.be/kjnJQSUwPBQ
Find the 4-point DFT of the sequence x[n]= {1, 0, 0, 1} using matrix method and verify the answer
9
by taking the 4-point IDFT of the result. https://youtu.be/JWlJfTaqjyY
Compute N point DFT of a sequence

1 1 2𝜋 𝑁
𝑥(𝑛) = 2 + 2 𝑐𝑜𝑠 [ 𝑁 (𝑛 − 2 )]
10 https://youtu.be/GQRB5X3caS0

Properties of DFT
1. Linearity Property
11
2. Circular Time shift property
https://youtu.be/-QwAI13DnKc
1. Circular Frequency shift property
12
2. Time Reversal Property https://youtu.be/8zOP9Xtkt84
3. Circular Convolution in Time domain (Convolution property)
OR
13
Show that multiplication of the DFTs of two sequences is equivalent to the circular convolution of the
https://youtu.be/-OenjaGEeZo
two sequences in the time – domain.
4. Symmetry Properties
14 Case 1: Real valued sequence https://youtu.be/mC8TLt-NRGg
Case 2: DFT of Real Even and Real Odd Sequences
15 Case-3: Both x(n) and its DFT X(k) are complex Valued https://youtu.be/pqEsW9S_oEU
5. Periodicity Property https://youtu.be/1jd8flGoSFY
16
 For the sequence x(n)= {-1,2,3,0,-4,1,2,-3}, calculate, without computing DFT.
7 7

∑ 𝑋(𝑘) 𝑎𝑛𝑑 ∑ |X(k)|2

17
𝑘=0 k=0
https://youtu.be/GcUI1AZrxrw

18 Given x(n)= {1, 2, 3, 4}, find y(n), if y(k)= x((k-2))4 https://youtu.be/6s6YOs6DIe8

19 Concept of Circular time shift https://youtu.be/Cbuq9Lhv9-U
Circular Convolution of two discrete sequences
20
• Using Time domain (formula) approach https://youtu.be/MeY745oUHH0
21 • Using Concentric circle method https://youtu.be/wIqlpV2wadc
22 • Using Frequency domain approach https://youtu.be/SaBsba1g5pQ
23 Circular Convolution using Matrix Multiplication method https://youtu.be/G9HFlSLUlv4
3𝜋𝑛 𝜋𝑛
Compute DFT of the sequence 𝑥(𝑛) = 𝑠𝑖𝑛 ( ) + 𝑐𝑜𝑠 ( 4 ), 0≤ 𝑛 ≤ 3,
4
https://youtu.be/SacgBWyE6j4
using linearity property of DFT
The 4 point DFT of a length-4 sequence 𝑥(𝑛)is given by X(𝑘) = {8, -1+j, -2, -1-j}. Obtain 𝑌(𝑘), the
24 −𝑗𝜋𝑛 https://youtu.be/J8dz5mhi_tY
4 point DFT of the sequence 𝑦(𝑛) = 𝑒 2 𝑥((𝑛 − 1))
Determine the 4 - point circular convolution of the sequences
2𝜋𝑛 2𝜋𝑛
𝑥1 (𝑛) = 𝑐𝑜𝑠 ( ) 𝑥2 (𝑛) = 𝑠𝑖𝑛 ( ) https://youtu.be/lANZ44ab5Ns
𝑁 𝑁
25 Using the time – domain formula.

Verify the result using frequency domain approach using DFT and IDFT https://youtu.be/V5WZNmpl_U4
Compute the N – point DFT of the following signals.
26
(𝑖) 𝑥(𝑛)= 𝛿(𝑛−𝑛0); 0 <𝑛0<𝑁 (𝑖𝑖) 𝑥(𝑛)=𝑎𝑛 ; 0≤𝑛≤(𝑁−1 https://youtu.be/12dG9u50HCw
Compute N point DFT of a sequence

1 1 2𝜋 𝑁
𝑥(𝑛) = 2 + 2 𝑐𝑜𝑠 [ 𝑁 (𝑛 − 2 )]
27 https://youtu.be/GQRB5X3caS0
 Compute the circular convolution using DFT and IDFT method for the following sequences
28
x1 (n) = {1,2,3,1} and x2 (n) = {4,3,2,2} https://youtu.be/7aFkjvC3R0M
Let x(n) be a real sequence of length N and its N-point DFT is X(k), show that i) X(N − K) =
X ∗ (K) ii) X(0) is real
29 https://youtu.be/7xuHHqUiG5s
N
iii) If N is even, then X ( 2 ) is real

Compute the circular convolution of the following sequences using DFT and IDFT method.
30
𝑥1 (𝑛) = {1,2,3,4} 𝑥2 (𝑛) = {4,3,2,1} https://youtu.be/0_rDSECGJ80
1 1 2𝜋 𝑁
If 𝑤(𝑛) = 2 + 2 𝑐𝑜𝑠 [ 𝑁 (𝑛 − 2 )] what is the DFT of the window sequence
31 https://youtu.be/uuFn5c3VSMA
𝑦[𝑛] = 𝑥[𝑛]. 𝑤[𝑛]? Relate the answer in terms of X(k)
Compute 4-point DFT of sequence x(n)= {1,2,3,4}. Using time shift property find the DFT Y(K), if y(n)=
32
𝑥((𝑛 − 3))4 https://youtu.be/zvzhhvzIyWI
The 4-point DFT of a sequence x(n) is given by X(k)={16, -4+j4, -4, -4-j4}. Determine
33
the energy of x(n) using Parseval’s theorem https://youtu.be/PvNyIsEq2R8

Linear Filtering methods based on DFT

(overlap-save method and overlap-Add method)
Linear Filtering using DFT.
1
understand how to perform Linear convolution using circular convolution. https://youtu.be/pjYDQZLMhZo
By means of DFT & IDFT, determine the response of the FIR filter with impulse
2 https://youtu.be/Pbmyr3Ae4Ng
response ℎ(𝑛) = {5, 6, 7} to the input sequence 𝑥(𝑛) = {1, 2, −1, 5 6}.
3 Overlap and save method explanation with example https://youtu.be/XsHCM2l4pp4
4 Overlap and add method explanation with example https://youtu.be/xNZPv52sZtk
Using overlap & save method, compute the output of an FIR filter with impulse response
5 ℎ(𝑛) = {1, 2, 3} and input 𝑥(𝑛) = {2, −2, 8, −2, −2, −3, −2, 1, −1, 9, 1, 3} use only 8 –point circular https://youtu.be/XsHCM2l4pp4
convolution in your approach.
 Using overlap & save method, find the output of an FIR filter whose impulse response ℎ(𝑛) = {1, −
6 2, 3} and input 𝑥(𝑛) = {2, 3, -1, 0, 5, 2, -3, 1}. https://youtu.be/GFEcPZWbvqU
use only 6-point circular convolution.
Using overlap & add method, compute the output of an FIR filter with impulse response
ℎ(𝑛) = {1, − 2, 3} and input 𝑥(𝑛) = {2, −2, 8, −2, −2, −3, −2, 1, −1, 9, 1, 3} use only 8– point circular
7 https://youtu.be/xNZPv52sZtk
convolution in your approach.
Find the output y(n) of a filter whose impulse response is h(n) = {1, 1, 1} and the input signal x(n) =
8
{3, -1,0,1,3,2,0,1,2,1} using overlap-add method. Assume the length of each block N is 6. https://youtu.be/vhNd9hbbjRE
Find the output y(n) of the filter whose impulse response is h(n)={1, 2} and the input signal to the
9 filter is x(n)={1,4,3,2,7,4-7,-7,-1,3,4,3} using overlap save method. Use only 5 point circular https://youtu.be/_ln8bZzyqN8
convolution approach
Find the response of an LTI system with an impulse response is h(n)={3, 2,1} for the input
10
x(n)={2,-1,-1,-2,-3,5,6,-1,2,0,2,1} using overlap add method use 8 point circular convolution https://youtu.be/yVkX9lscZxE
Using overlap add method, compute the output of an filter with an impulse response h(n)={1,- 2,3}
11
and input x(n)={1,0,2,0,-1,-2,3,-3,1,2} use 8 point circular convolution https://youtu.be/Pm-4Yw_RqKQ

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 MODULE 3
Efficient Computation of the DFT-
FFT Algorithms:
Direct Computation of the DFT,
Radix-2 FFT Algorithms:
computation of DFT and IDFT in
decimation in time.
 Module-3- Fast-Fourier-Transform (FFT) algorithms
1 Develop radix – 2, DIT – FFT algorithm and write signal flow graph for 𝑁 = 8. https://youtu.be/9WNJflDOk14
Find the 8-point DFT of the sequence
2 𝑥(𝑛) = {-1, 0, 2, 3, -4, -2, 0, 5} using Radix-2 DIT – FFT algorithm.
https://youtu.be/eDYNIEBvz9A

3
Given X(k) = {1, j4, 1, -j4}, find x(n) using Radix-2 DIT – FFT algorithm. https://youtu.be/4-mTQDodu4I
What do you mean by computational complexity? Compare the direct computation
and FFT algorithms. In the direct computation of 32-point DFT of x(n), How
many
4
i) Complex multiplications (ii) Complex additions
https://youtu.be/qaBkK-q6i3w
ii) Real multiplications. iv) Real Additions
v) Trigonometric function evaluations are required.
5 Develop 8-point DIT-FFT Radix-2 algorithm and draw the signal flow graph. https://youtu.be/9WNJflDOk14
Using DIT-FFT algorithm compute the DFT of a sequence x[n]=[1,1,1,1,0,0,0,0]
6 https://youtu.be/RHXvrzH0XvA
7 Given X(k)= [0, j4, 0, -j4]. Find x(n) using Radix 2 DIT FFT algorithm https://youtu.be/fdAyK_5vaYc
Comparison between computation of DFT using direct method and using FFT
8
algorithms. https://youtu.be/aoez3R7Y2gg
9 Given 𝑥(𝑛) = (𝑛 + 1) 𝑓𝑜𝑟 0 ≤ 𝑛 ≤ 7. Find 𝑋(𝑘) using DIT – FFT algorithm. https://youtu.be/smDROo5md-k
Compare the Complex multiplications and additions for the direct computation of
10
DFT versus FFT algorithm for N-128. https://youtu.be/qaBkK-q6i3w

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