Chapter 8 the discrete fourier transform 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) 8.1 representation of periodic sequences: the discrete fourier series 1 ~ x[ n] N ~ X [k ] kn WN e N 1 ~ kn X [ k ]WN (1), n 0,..N 1 k 0 N 1 ~ x[n]WN (2), k 0,..N 1 kn n 0 j 2kn / N ~ X [ k rN ] ( k rN ) n ~ ~ X [k ] x [n]W N n 0 N 1 EXAMPLE. Figure 8.1 ~ X [k ] 4 n 0 kn W10 e N=10 j 4k / 10 sin(k / 2) sin(k / 10 ) phase X denotes：magnitude=0，phase is indeterminate Figure 8.2 8.2 the fourier transform of periodic signals 2 ~ N 1 2 ~ 2k ~ j X [k ] X (e ) X [k ] ( ) N N N k 0 0 2 N other 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. k FIGURE 8.5 8.3 properties of the discrete fourier series DFS ~ DFS ~ DFS ~ ~ ~ ~ x [ n] X [k ], x1[n] X 1[k ], x2 [n] X 2 [k ] DFS ~ ~ ~ ~ 1.linearity : ax1[n] bx2 [n] aX 1[k ] bX 2 [k ] N=4，12 points DFS two periodic sequences with different period N=6，12 points DFS both period=12 compositive sequence N=12，12 points DFS 2.shift of DFS km ~ ~ a sequence : x [n m] W X [k ] N nl ~ DFS ~ W x [ n] X [k l ] N DFS ~ ~ 3.duality : X [n] Nx [k ] 4.symmetry properties: DFS ~ DFS ~ ~ ~ x * [n] X * [k], x * [n] X * [k] DFS 1 ~ 1 ~ ~ ~ ~ ~ Re{ x[n]} (x[n] x * [n]) (X[k] X * [k]) X e [k] 2 2 DFS 1 ~ 1 ~ ~ ~ ~ ~ jIm{ x[n]} (x[n] x * [n]) (X[k] X *[k]) X o [k] 2 2 DFS 1 ~ 1 ~ ~ ~ ~ ~ x e [n] (x[n] x *[n]) (X[k] X *[k]) Re{ X[k ]} 2 2 DFS 1 ~ 1 ~ ~ ~ ~ ~ xo [n] ( x [n] x * [n]) ( X [k ] X * [k ]) j Im{ X [k ]} 2 2 For a real sequence: ~ ~ x[n] x*[n] ~ ~ X [k ] X *[k ] ~ ~ Re{ X [ k ]} Re{ X [ k ]} ~ ~ Im{ X [ k ]} Im{ X [ k ]} ~ ~ | X [ k ] || X [ k ] | ~ ~ X [k ] X [k ] EXAMPLE. DFS of real sequence FIGURE 8.2 5. periodic convolution : N 1 ~ x3 [n] ~ x1[n] ~ x2 [ n ] ~ x1[m]~ x2 [ n m ] ~ x2 [ n ] ~ x1[n] m 0 Periods of 3 sequences are all N. (1)if : ~ x3 [n] ~ x1[n] ~ x2 [n] ~ ~ ~ then : X 3 [k ] X 1[k ] X 2 [k ] (2)if : ~ x3 [n] ~ x1[n]~ x2 [n] ~ ~ then : X 3 [k ] 1 N ~ X [k ] X 2 [k ] 1 graphic method to calculate periodic convolution Figure 8.3 8.4 fourier representation of finite-duration sequences: Definition of the discrete fourier transform ~ x [ n] x[n rN ] x[n mod N ] x[(( n)) N ] r ~ x[n] x[n]RN [n] EXAMPLE. Figure 8.8 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 N 1 kn X [k ] x[ n]WN , k 0,1,....N 1 n 0 N 1 1 kn x [ n ] X [ k ] W , n 0,1,...N N N k 0 N 1 X [0] x[ n] n 0 1 N 1 x[0] X [k ] N k 0 X [ k ] X (e j )| 2 N X ( z) | k z e j 2 N k EXAMPLE. Figure 8.10 periodic extension with period 5 x[(( n)) 5 ] DFS{~ x [n]} ~ X [ k ]R5 [ k ] 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 X (e j )| 2 k M then the result of IDFT is ： kn x[ n]WM , k 0,1...M 1 n x'[n] ( x[ n rM ]) RN [n] r 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： X (e j ) 1 e jN N 1 / N X [k ] k k 0 1 WN e j prove : X (e j N 1 ) x[ n]e j n n 0 1 N N 1 N 1 N 1 ( n 0 X [k ](WN e k 0 1 e jN kn jn 1 N ) n 0 N 1 / N k 0 X [k ] k 1 WN e j N 1 X [k ]WN k 0 kn )e j n 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： M points IDFT X [k ] 2 M点取样 FT x1[n] X1 (e j x2 [n] x[n rM ] RM [n] r 取长为M 的主周期 ) M点IDFS ~ ~ M点取样 X 3[k ] x3[n] x[n rM ] r periodic extension in time domainsampling in frequency domain summary 8.5 1. 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 x[n] DFT X [k ], x1[n] DFT 1.linearity : ax1[n] bx2 [n] X 1[k ], x2 [n] DFT DFT aX1[k ] bX 2 [k ] 2.circular shift of a sequence x1[n] x[(( n m)) N ]RN [n] x[(( n ( N m))) N ]RN [n] x[((n m)) N ]RN [n] W ln N x[n] DFT DFT W km X [k ] N X [((k l )) N ]RN [k ] X 2 [k ] EXAMPLE. circular shift Figure 8.12 EXAMPLE. （8.42） | H1[k ] | 8points DFT| H [k ] | 2 h1[n] h2 [n] | H1 (e j ) | 1024 points DFT | H 2 (e j )| 3.duality : X [n] DFT Nx[((k )) N ]RN [k ] Nx[ N k ] EXAMPLE. x[n] n cos(0.2n) | X [k ] || DFT {x[n]} | DFT { X [n]} X '[k ] X [(( k )) N ]RN [k ] X [(( N k )) N ]RN [k ] X [ N k ] X [k ] X '[k ] 4. Symmetry properties: * x [ n] DFT * * * X [((k )) N ]RN [k ] X [(( N k )) N ]RN [k ] X [ N k ] * * x [((n)) N ]RN [n] x [ N n] Re{ x[n]} 1 ( x[n] x * [n]) DFT DFT 1 * X [k ] ( X [k ] X * [ N k ]) X ep [k ] 2 2 DFT 1 1 ~ ~ ~ j Im{x [n]} ( x [n] x * [n]) ( X [k ] X * [ N k ]) X op[k ] 2 2 xep [n] 1 xop[n] 1 ( x[n] x * [ N n]) DFT 1 2 2 ( x[n] x * [ N n]) ( X [k ] X * [k ]) Re{ X [k ]} 2 DFT 1 2 ( X [k ] X * [k ]) j Im{ X [k ]} definition ： X ep [ k ] : periodic conjugate - symmetric components X op [ k ] : periodic conjugate - antisymmet ric components sequence with length N can be decomposed ： X [ k ] X ep [ k ] X op [ k ] while ： X ep [ k ] 1 2 ( X [ k ] X * [ N k ]) X ep * [ N k ]，length N real part is even symmetric , imaginary part is odd symmetric X op [ k ] 1 2 ( X [ k ] X * [ N k ]) X op * [ N k ]，lenth N real part is odd symmetric , imaginary part is even symmetric For a real sequence: x[n] x * [n] X [k ] X * [ N k ] Re{ X [k ]} Re{ X [ N k ]} Im{ X [k ]} Im{ X [ N k ]} | X [k ] || X [ N k ] | X [k ] X [ N k ] EXAMPLE. x[n] 0.5n cos(0.5n), n 0...9 Real{X[k]} |X[k]| N=10 DFT of real sequence Imag{X[k]} arg{X[k]} N=9 |X[k]| Arg{X[k]} 5.circular convolution y[n] x1[n]( N ) x2[n] ( x1[((n)) N ] x2[((n)) N ])RN [n] N 1 ( x [((m)) 1 N ] x2 [((n m)) N ]) RN [ n] m 0 Length of x1[n],x2[n],y[n] are N. (1) x1[n]( N ) x2 [n] (2) x1[n]x2 [n] DFT DFT 1 N X 1[ k ] X 2 [ k ] X 1[k ]( N ) X 2 [k ] EXAMPLE. graphic method to calculate circular convolution x[n]( N ) [n 1] x[(( n 1)) N ]RN [n] Figure 8.14 convolution result depends on N EXAMPLE. Figure 8.15 N=L Figure 8.16 N=2L EXAMPLE. 6.paswal’s theory N 1 N 1 x[n] y * [n] N X [k ]Y * [k ] 1 n 0 k 0 N 1 n 0 2 | x[ n] | 1 N N 1 k 0 2 | X [k ] | 8.7 linear convolution using the discrete fourier transform if : y[n] x[n] * h[n] then : x[n（ ] N）h[n] y[n rN ]R N [ n] r PROVE x[n] h[n] y[n] FT X (e j ) H (e j ) Y (e j ) N point sample according to properties of circular convolution X [k ]H [k ] IDFT x[n]( N )h[n] Y [k ] according to spectral sampling y[n rN ]R r If N>=N1+N2-1,then x[n]*h[n]=x[n](N)h[n] N [ n] EXAMPLE. Figure 8.18 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: (1)calculate N point circular convolution by linear convolution (a) y[n] x[n] * h[n] (b) x[ n]( N )h[n] y[n rN ]R N [ n] r (2) calculate linear convolution by circular convolution (a) zero padding x[n] and h[n] to length of N L1 L2 1 (b) x[n] h[n] x[n]( N )h[n] (3) calculate linear convolution by DFT (a) zero padding x[n] and h[n] to length of (b) N po int s DFT of x[n] and h[n] (c) x[n]( N )h[n] IDFT { X [k ]H [k ]} x[n] * h[n] x[n]( N )h[n] N L1 L2 1 implementing linear time-invariant FIR systems using the DFT Figure 8.22 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 length of h[n] is P (1)segment x(n) into sections of length L; (2)fill 0 into h(n) and some section of x(n) , then do L+P-1 points FFT ; (3) calculate y (n) IFFT {H (k ) X (k )}，n 0,...L P 2 (4)add the points n=0…P-2 in y (n) to the last P-1 points in the former section y[n]， the output for this section is the points n=0…L-1 y0 (n) 0 h[0] 0 h[1] 0 h[2] x[n 3]h[3] ... n L,...L P 2 y1 (n) x[n] h[0] x[n 1] h[1] x[n 2] h[2] 0 h[3] 0 h[4] ..., n 0,..P 2 y2 (n) y1[n] y2 [n], n 0,...P 2 P-1 points overlap-save method the length of h[n] is P (1)segment x(n) into sections of length L, overlap P-1 points; (2)fill 0 into h(n) and some section of x(n) , then do L points FFT ; (3) calculate y(n) IFFT{H (k ) X (k )}，n 0,...L 1 (4) the output for this section is the L-P+1 points n=P-1,…L-1 of y[n] 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。 Figure 8.24 P-1 points linear convolution result 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：X [ k ] x[ n] DCT 2：X [ k ] N 1 ck 2 N cn N 1 ck X [ k ] cos( n 0 N 1 ck kn N 1 N 1 2 cn x[ n] cos( n 0 2 N x[ n] N 1 2 x[ n] cos( N 1 n 0 kn N 1 k ( 2n 1) n 0 ck X [ k ] cos( ),0 k N 1 ),0 n N 1 ),0 k N 1 2N k (2n 1) 2N ),0 n N 1 DCT 3：X [ k ] 2 N x[ n] 2 N DCT 4：X [ k ] N 1 N 1 ck X [ k ] cos( n 0 N 1 x[ n] cos( x[ n] n 0 1 / 2, k 0 ck 1,1 k N 1 n( 2k 1) ),0 n N 1 ( 2n 1)(2k 1) ),0 k N 1 4N N 1 N ),0 k N 1 2N 2N n 0 2 n( 2k 1) n 0 N 2 cn x[ n] cos( X [ k ] cos( ( 2n 1)(2k 1) 4N ),0 n N 1 symmetric and periodic extension of signal, then do DFS and get DCT by taking the dominant period。 DCT-1 -2 -2 -2–1. –1. –1.0. 0. 0.1. 1. 1.2. 2. 2.3. 3. 3.4.4. 4.555 DCT-2 -4 -4 -4–3 –3 –3-2-1 -2-1 -2-10. 0.0.1. 1.1.2. 2.2.3. 3.3.4. 4.4.5. 5.5.6. 6.6.777 relationship between 2Npoinsts DFT of extended sequence and N-points DCT of original sequence DCT 2 x[ n], n 0,...N 1 y[ n] X [ 2 N 1 n], n N ,...2 N 1 2 N 1 Y [k ] N 1 kn y[ n]W2 N n 0 kn x[ n]W2 N k ( 2 N 1 n ) x[ n]W2 N n 0 N 1 k / 2 2W2 N x[ n] cos( ( 2n 1)k 2N n 0 k / 2 2W2 N X [ k ] /( 2 N ck ) n N N 1 n 0 kn x[ n]W2 N n 0 N 1 2 N 1 ) kn x[ 2 N 1 n]W2 N Compare with DFT:energy compaction property DCT 012 345 67 DFT 012 345 67 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 8.26 8.29 8.39 8.45 8.49

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# Chapter 8