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 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 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 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 7.1.5 DFT Transformation Matrix The DFT can be represented in this way This introduces the widely-used and convenient notations: whence
7 7.1.6 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 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 7.2.2 DFT Properties - II –Circular Shift: –Duality: –Parseval’s Theorem (DFT conserves energy) –Cyclic (circular convolution): If then
10 7.2.3 DFT Properties – Circular Shift Example
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 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
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 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 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.