# CE 40763 Digital Signal Processing Fall 1992 Discrete Fourier Transform (DFT)

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CE 40763Digital Signal Processing

Fall 1992

Discrete Fourier Transform (DFT)Hossein SametiDepartment of Computer Engineering Sharif University of Technology

12MotivationThe DTFT is defined using an infinite sum over a discrete time signal and yields a continuous function X()not very useful because the outcome cannot be stored on a PC.Now introduce the Discrete Fourier Transform (DFT), which is discrete and can be stored on a PC.We will show that the DFT yields a sampled version of the DTFT.

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 2Review of Transforms3

C.

D.C.C.

D.ComplexInf. orFinite

D.Int.periodicperiodic

D.Int.finitefinite3Review of Transforms4

4Discrete Fourier Series (DFS)5

Decompose in terms of complex exponentials that are periodic with period N.

How many exponentials?N

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 5Discrete Fourier Series (DFS)6

Exponentials that are periodic with period N.

arbitrary integer* Proof:

1

67Discrete Fourier Series (DFS)

How to find X(k)?Answer:

Proof : substitute X(k) in the first equation.It can also easily be shown that X(k) is periodic with period N:

arbitrary integerHossein Sameti, Dept. of Computer Eng., Sharif University of Technology 7DFS Pairs8Analysis:

Synthesis:

Periodic N pt.seq. in time domainPeriodic N pt.seq. in freq. domainHossein Sameti, Dept. of Computer Eng., Sharif University of Technology 8Example9

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 9Example (cont.)10

(eq.1)(eq.2)(eq.1) & (eq.2)

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 10Shift property:

Periodic convolution:Properties of DFS11

Period N

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 11Periodic convolution - Example12

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 12In the list of properties:

Properties of DFS13

Where:Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology Properties of DFS14

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 14Properties of DFS15

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 15Discrete Fourier Transform16

N pt.N pt.DFTN pt.N pt.DFTDTFTDFSHossein Sameti, Dept. of Computer Eng., Sharif University of Technology 161) Start with a finite-length seq. x(n) with N points (n=0,1,, N-1).2) Make x(n) periodic with period N to get Deriving DFT from DFS17

Extracts one period of

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 173) Take DFS of4) Take one period of to get DFT of x(n): 18Deriving DFT from DFS (cont.)

N pt.N pt.N pt.periodicN pt.periodicHossein Sameti, Dept. of Computer Eng., Sharif University of Technology 18Example19

19Definition of DFT:Discrete Fourier Transform20

N pt. DFT of x(n)

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 20Example21

21Example, contdMehrdad Fatourechi, Electrical and Computer Engineering, University of British Columbia, Summer 201122

22Relationship between DFT and DTFT23

DFT thus consists of equally-spaced samples of DTFT.

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 2324Relationship between DFT and DTFT

8 pt. sequence8 pt. DFT

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 24M pt. DFT of N pt. Signal25

So far we calculated the N pt. DFT of a seq. x(n) with N non-zero values:Suppose we pad this N pt. seq. with (M-N) zeros to get a sequence with length M.

We can now take an M-pt. DFT of the signal x(n)Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 2526M pt. DFT of N pt. Signal

DFTN pt.M pt.Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 26Example274 pt. DFT:

6 pt. DFT:

8 pt. DFT:

100 pt. DFT:

How are these related to each other?2728M pt. DFT of N pt. Signal

Going from N pt. to 2N pt. DFTHossein Sameti, Dept. of Computer Eng., Sharif University of Technology 2829M pt. DFT of N pt. SignalN pt.DFTN pt. seq.N pt.2N pt.DFTN pt. seq.padded with N zeros2N pt.What is the minimum number of N needed to recover x(n)?Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 29Assume y(n) is a signal of finite or infinite extent.

Problem Statement30

Sample at N equally-spaced points.

N pt. sequence.

N pt. sequence.What is the relationship between x(n) and y(n)?What happens if N is larger , equal or less than the length of y(n)?Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 30We start with x(n) and find its relationship with y(n):Solution to the Problem Statement31

Change the order of summation:

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 3132Solution to the Problem StatementHowever, we have shown that:

Convolution with train of delta functions

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 3233Solution to the Problem Statement

One period of the replicated version of y(n) ExamplesIf we sample at a sampling rate that is higher than the number of points in y(n), we should be able to recover y(n).

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 33Shift property:Properties of DFT34

N pt. seq.

The above relationship is not correct, because of the definition of DFT.The signal should only be non-zero for the first N points.

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 34In the list of properties:

Properties of DFT35

where:

andwhere:Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 36Summery of Properties of DFT

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 3637Summery of Properties of DFT

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 37Using DFT to calculate linear convolution38Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 38We are familiar with, linear convolution.Question: Can DFT be used for calculating the linear convolution?The answer is: NO! (at least not in its current format)We now examine how DFT can be applied in order to calculate linear convolution. Convolution39Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 39Definitions of convolution40

Linear convolution:

Application in the analysis of LTI systems Periodic convolution:A seq. with period NHossein Sameti, Dept. of Computer Eng., Sharif University of Technology 40Definitions of convolution(cont.)41

Circular convolution:N pt. seq. Circular convolution is closely related to periodic convolution.

N pt. DFT of x1N pt. DFT of x2N pt. DFT of x3Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 41Example: Circular Convolution42

Circular convolution?Make an N pt. seq.

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 4243Example: Circular Convolution

43Circular Convolution & DFT44

We know from DFS properties:

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 4445Circular Convolution & DFT

If we multiply the DFTs of two N pt. sequences, we get the DFT of their circular convolution and not the DFT of their linear convolution.Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 45Example : Circular Convolution46

Calculate N pt. circular convolution of x1 and x2 for the following two cases of N:N=LN=2LHossein Sameti, Dept. of Computer Eng., Sharif University of Technology 4647Case 1: N=L

N pt. DFT of x1

IDFTHossein Sameti, Dept. of Computer Eng., Sharif University of Technology 4748Case 1: N=L

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 4849Case 2: N=2L Pad each signal with L extra zeros to get an 2L pt. seq.:

N=2L pt. DFT of x1

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 4950Case 2: N=2L

Same as linear convolution!!50Our hypothesis is that if we pad two DT signals with enough zeros so that its length becomes N, we can use DFT to calculate linear convolution.Using DFT to Calculate linear Convolution51

L pt. seq.

P pt. seq.

Using DFTGoal: calculateHossein Sameti, Dept. of Computer Eng., Sharif University of Technology 5152

Using DFT to Calculate linear Convolution

To get DFT, we have to sample the above DTFT at N equally-spaced points:

Solution to the problem statement(Eq.1)Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 5253Using DFT to Calculate linear Convolution

N pt. DFT of x1N pt. DFT of x2

(Eq.2)

Circular convolution(Eq.1)(Eq.2)

Replicated version of the linear convolutionOn the other hand, we know that:Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 53In other words, the N pt. circular convolution of two DT signals is the same as their linear convolution, if we make the result of linear convolution periodic with period N and extract one period. 54Using DFT to Calculate linear Convolution

To avoid aliasing:We can thus use DFT in order to calculate the linear convolution of two sequences!Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 541) Start with Algorithm for calculating linear convolution using DFT55

L pt. seq.

P pt. seq.3) Pad with N-L zeros to get N points.4) Pad with N-P zeros to get N points.

5) Calculate the N pt. DFTs of the above two sequences and multiply them together.6) Calculate IDFT of the resulting N pt. sequence.2) Choose

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 55Using DFT for linear filtering when one sequence is very long56h(n)

A very long sequenceAn FIR filter with a limited number of taps (P) Examples of this situation?Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 5657Using DFT for linear filtering when one sequence is very longSolution to the problemOverlap - addMain idea: using the following property:

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 57Segment the long sequence into non-overlapping chunks of data with the length of L.Convolve each chunk with h(n) to get (L+P-1) new points.Add the results of the convolution of all chunks to get the final answer.Overlap- add method58Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 5859Overlap- add method- Example

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 5960Overlap- add method- Example

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 6061Overlap- add method- Example

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 61Discussed DFS and DFT and examined the relationship between DFT and DTFTShowed how DFT can be used for calculating convolution sum.Next: Fast Fourier TransformSummary62Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology 62