1. 1 7.1 Discrete Fourier Transform (DFT) 7.2 DFT Properties 7.3 Cyclic Convolution 7.4 Linear Convolution via DFT Chapter 7 Discrete Fourier Transform Section 8.1-8.7

2. 2 Definition: The Discrete Fourier Transform (DFT) of the finite length sequence is Definition: The Inverse Discrete Fourier Transform (IDFT) of is given by The following notation will be used: 7.1.1 Discrete Fourier Transform (DFT)

3. 3 7.1.2 Discrete Fourier Series (DFS) Periodic Extension: Given a finite-length sequence define the periodic sequence by The sequence with period N is called the periodic extension of x[n]. It has a fundamental frequency. does not have a Z-transform or a convergent Fourier sum (why?). But it does have a DFS representation. It is actually the DFS that is the true frequency representation of discrete periodic signals. The DFT is just one period of the DFS.

4. 4 7.1.3 DFT and DFS DFS analysis and synthesis pair is expressed as follows: Practical significance: –The length-N DFT of the length-N signal contains all the information about. It is convenient to work with. –Whenever the DFT is used, actually the DFS is being used – computations involving are affected by the true periodicity of the coefficients.

5. 5 7.1.4 Relation with Other Transforms The DFT samples the Z-transform at evenly spaced samples of the unit circle over one revolution: In other words, the DFT samples on period of the Fourier Transform at N evenly spaced frequencies

6. 6 7.1.5 DFT Transformation Matrix The DFT can be represented in this way This introduces the widely-used and convenient notations: whence

7. 7 7.1.6 DFT Transformation Matrix: Example The DFT matrices of dimension 2, 3, 4 are as follows: If we compute X by as follows: Where we observe that the real part of X[k] is even-symmetric, and the imaginary part is odd-symmetric – the DFT of the real signal.

8. 8 Let be length-N sequences indexed n=0,…,N-1. DFT Properties: –Linearity: For constant a, b: –Even Sequences: If x[n] is even: –Odd Sequences: If x[n] is odd: –Real Sequences: If x[n] is real: 7.2.1 DFT Properties - I

9. 9 7.2.2 DFT Properties - II –Circular Shift: –Duality: –Parseval’s Theorem (DFT conserves energy) –Cyclic (circular convolution): If then

10. 10 7.2.3 DFT Properties – Circular Shift Example

11. 11 The cyclic convolution is not the same as the linear convolution of linear system theory. It is a by-product of the periodicity of DFS/DFT. When the DFT X[k] is used, the periodic interpretation of the signal x[n] is implicit: if then for any integer m: Thus, just as the DFT X[k] is implicitly period-N (i.e., is the DFS), the inverse DFT is also implicitly period-N — the periodic extension of x[n]. 7.3.1 Cyclic Convolution – What?

12. 12 7.3.2 Cyclic Convolution – Why? Why is cyclic convolution not true linear convolution? Because a wraparound effect occurs at the “ends”: The procedure of each pair are summed around the circle. In a while, it will be seen that can be computed using

13. 13 7.3.3 Cyclic Convolution – Example 1

14. 14 7.3.4 Cyclic Convolution – Example 2

15. 15 7.4.1 Linear Convolution by DFT Of course, linear convolution is desired. Fortunately, the linear convolution can be computed via the DFT, with a minor modification. Method: To compute the linear convolution of a sequence x[n] of length- N 1 and a sequence h[n] of length- N 2. via the DFT, form the length N 1 + N 2 -1 zero-padded sequences and then

16. 16 7.4.2 Cyclic Convolution Example -1 The linear convolution is computed as the time instants (in this example) 0  n  4. This can be regarded as a form of time-aliasing – resulting from the sampling of the Fourier Transform.

17. 17 7.4.2 Cyclic Convolution Example -1 The linear convolution is computed as the time instants (in this example) 0  n  9. Aliasing is eliminated, so the result is the same as the linear convolution of the non-extended sequence.

18. 18 7.4.3 Cyclic Convolution Example -2