8.1 representation of periodic sequences:the discrete fourier series 8.2 the fourier transform of periodic signals 8.3 properties of the discrete fourier series 8.4 fourier representation of finite-duration sequences: Definition of the discrete fourier transform 8.5 sampling the fourier transform(point of sampling) 8.6 properties of the fourier transform 8.7 linear convolution using the discrete fourier transform 8.8 the discrete cosine transform(DCT) Chapter 8 the discrete fourier transform

8.1 representation of periodic sequences: the discrete fourier series

Figure 8.1 EXAMPLE. N=10

Figure 8.2 phase X denotes ： magnitude=0 ， phase is indeterminate

8.2 the fourier transform of periodic signals dispersion in time domain results in periodicity in frequency domain; Periodicity in time domain results in dispersion in frequency domain. DFS is a method to calculate frequency spectrum of periodic signals.

FIGURE 8.5

8.3 properties of the discrete fourier series N=4 ， 12 points DFS N=6 ， 12 points DFS compositive sequence N=12 ， 12 points DFS two periodic sequences with different period both period=12

For a real sequence:

FIGURE 8.2 EXAMPLE. DFS of real sequence

Periods of 3 sequences are all N.

Figure 8.3 graphic method to calculate periodic convolution

8.4 fourier representation of finite-duration sequences: Definition of the discrete fourier transform Figure 8.8 EXAMPLE. The last two expressions are only suitable to no aliasing.

Two derivations of definition: 1. Periodic extension of the finite-duration sequence with period N ; DFS of the periodic sequence ; DFT is the dominant period of DFS. 2. DTFT of the finite-duration sequence; DFT is the N-points spectral sampling.

duration of sequence is N

Figure 8.10 EXAMPLE. periodic extension with period 5

explanations ： 1. DFT and DFS have the same expression, but DFT are samples of frequency spectrum of the finite-duration sequence, DFS is frequency spectrum of periodic sequence 。 2. the periods of DFS in time and frequency domain is N, DFT in frequency domain is defined to be finite duration, but has the immanent period N 。 3. the meaning of DFT not only is samples of frequency spectrum, but also can reconstruct time-domain signal 。

8.5 sampling the fourier transform EXAMPLE. Figure 8.5 periodic extension with period 10 reflect frequency spectrum of signal more truly than figure 8.10

conclusion ： sequence with length N is extended to M by filling 0 in time domain, then do M-points DFT. We can get more dense samples of its FT, and can reconstruct time-domain signal by taking the first N nonzero values from the reconstructed signal 。 contrarily, if we want to get M-points samples of FT by DFT, we can use the method of filling 0 in time domain.

genetic instance ： Sequence with length N （ or infinite length ）, sample M points in frequency domain （ more than or less than or equal to N ）， then the reconstructed time-domain signal is dominant period of the periodic extension with period M of original signal （ maybe aliasing ）。 Viz. if then the result of IDFT is ： Conclusion ： when M<N, the reconstructed time-domain signal is domain period of the periodic extension with aliasing of original signal. Contrarily, if we want to get M (M<N) sample points of FT by DFT ， we can extend the sequence with period M in time domain, take the dominant period and do M-points DFT.

sampling theorem in frequency domain ： If sampling points N in frequency domain is more than the length of sequence ， the time-domain signal can be reconstructed ； and the sampling spectral line can be constructed to be continuous spectral function by ideal interpolation ：

M 点取样 DFT of a finite-duration sequence 、 DFS of a periodic sequence are both samples of FT of another sequence, then the relationship among the three sequences in time domain ： periodic extension in time domain  sampling in frequency domain 取长为 M 的主周期

summary DFT is N-points samples of frequency spectrum of sequence with length N. the more spectral sampling points, the more genuine to reflect the frequency spectrum 2. get M-points spectral samples of N-points sequence by M-points DFT: （ 1 ） M=N ， do M-points DFT directly （ 2 ） M>N ， extend x[n] to M points by filling 0 ， then do M-points DFT （ 3 ） M<N ， periodic extension of x[n] with period M and aliasing ， take the dominant period with length M, then do M-points DFT 3. whether spectral sampling can reconstruct original time-domain signal spectral sampling theorem ： if spectral sampling points is larger than or equal to the length of signal, the time-domain signal can be reconstructed. Contrarily, it can not be done. 4. If frequency spectrum is the same ， its samples are equal ； contrarily, it does not come into existence 。 if frequency spectrum has linear phase, its samples has linear phase, too ； contrarily, it does not come into existence 。

8.6 properties of the fourier transform 2.circular shift of a sequence

Figure 8.12 EXAMPLE. circular shift

（ 8.42 ） EXAMPLE. 8points DFT 1024 points DFT

EXAMPLE.

4. Symmetry properties:

For a real sequence:

N=10 EXAMPLE. DFT of real sequence Real{X[k]}Imag{X[k]} |X[k]|arg{X[k]}

N=9 |X[k]| Arg{X[k]}

5.circular convolution Length of x1[n],x2[n],y[n] are N.

Figure 8.14 EXAMPLE. graphic method to calculate circular convolution

Figure 8.15 N=L EXAMPLE. convolution result depends on N

Figure 8.16 N=2L EXAMPLE.

6.paswal’s theory

8.7 linear convolution using the discrete fourier transform

N point sample IDFT according to properties of circular convolution according to spectral sampling FT PROVE If N>=N1+N2-1,then x[n]*h[n]=x[n](N)h[n]

Figure 8.18 EXAMPLE. linear convolution shift right of the linear convolution shift right of the linear convolution 6 points circular convolution= linear convolution with aliasing 12 points circular convolution = linear convolution

Conclusion: (2) calculate linear convolution by circular convolution (1)calculate N point circular convolution by linear convolution (3) calculate linear convolution by DFT

Figure 8.22 implementing linear time-invariant FIR systems using the DFT consideration ： （ 1 ） deal with data when input them （ 2 ） operation speed （ 3 ） the counts of input and output data are equal the following two methods is used in the situation such that real-time and speedy operation is required ； time-domain method is commonly used 。

overlap-add method (1)segment into sections of length L; (2)fill 0 into and some section of, then do L+P-1 points FFT ; (3) calculate (4)add the points n=0…P-2 in to the last P-1 points in the former section y[n] ， the output for this section is the points n=0…L-1 length of h[n] is P P-1 points

Figure 8.24 overlap-save method the length of h[n] is P P-1 points linear convolution result If do L+P-1 points DFT, then wipe off the first and last P-1 points in the result, respectively, output is the middle L-P+1 points 。 To guarantee the output is linear convolution result, the minimum points of DFT is L 。 (1)segment into sections of length L, overlap P-1 points; (2)fill 0 into and some section of, then do L points FFT ; (3) calculate (4) the output for this section is the L-P+1 points n=P-1,…L-1 of y[n]

Conclusion : use of DFT ： （ 1 ） calculate spectral sample of signals （ 2 ） calculate sample of frequency response of systems （ 3 ） frequency-domain realization for FIR system

8.8 the discrete cosine transform(DCT)

DCT-1 DCT-2 symmetric and periodic extension of signal, then do DFS and get DCT by taking the dominant period 。

relationship between 2N- poinsts DFT of extended sequence and N-points DCT of original sequence

Compare with DFT:energy compaction property DCT DFT

summary 8.1 representation of periodic sequences: the discrete fourier series 8.2 the fourier transform of periodic signals 8.3 properties of the discrete fourier series 8.4 fourier representation of finite-duration sequences: Definition of the discrete fourier transform 8.5 sampling the fourier transform (point of sampling) 8.6 properties of the fourier transform 8.7 linear convolution using the discrete fourier transform 8.8 the discrete cosine transform (DCT)

requirements ： definition, calculation and properties of DFS; derivation of definition of DFT ： DFS or spectral sampling; concepts of spectral sampling, ， time-domain periodic extension; properties of DFT ： linearity 、 circular shift, symmetry, circular convolution 、 paswal’s theory; relationship between linear and circular convolution; derivation of definition DCT and comparison with DFT. key and difficulty ： spectral sampling and properties of DFT

exercises