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1、1,Chap 5 Finite-Length Discrete Transforms,The Discrete Fourier Transform Relation Between DTFT and DFT Operation on Finite-Length Sequences Classification of Finite-Length Sequences DFT Symmetry Relation Discrete Fourier Transform Theorems Computation of the DFT of Real Sequence Linear Convolution
2、Using the DFT,2,1. Definition,5.1 Orthogonal Transforms,-analysis equation,-synthesis equation,-basis sequence,Length-N,3,Condition:,5.1 Orthogonal Transforms,2. Parsevals relation,4,5.2 The Discrete Fourier Transform,1. Review,Time domain Frequency domain Continue aperiodic FT Aperiodic continue Co
3、ntinue periodic FS Aperiodic discrete Discrete aperiodic DTFT Periodic continue Discrete periodic DFS Periodic discrete,5,2. Make a signal discrete and periodic,The engineering signals are often continuous and aperiodic. If we want to process the signals with DFT, we have to make the signals discret
4、e and periodic.,1) Sampling to make the signal discrete,2) Make the signal periodic:,a) If xn is a limited length N-point sequence, see it as one period of a periodic signal that means extend it to a periodic,b) If xn is an infinite length sequence, cut-off its tail to make a N-point sequence, then
5、do the periodic extending. The tail cutting-off will introduce distortion. We must develop truncation algorithm to reduce the error, which is windowing.,6,2. Make a signal discrete and periodic,7,5.2.1 Definition,1. Definition,-discrete Fourier transform,basis sequence,-complex exponential sequence,
6、Suppose: xn,Xk-finite-length sequence, N- length; i.e., xn=0 when nN-1,8,basis sequence has the orthogonality property,5.2.1 Definition,Proof:,9,1) Xk is also a length-N sequence in the frequency domain 2) Xk is called the discrete Fourier transform (DFT) of the sequence xn 3) Using the notation WN=
7、e-j2 /N the DFT is usually expressed as:,2. Note,5.2.1 Definition,10,3. IDFT,1) The inverse discrete Fourier transform (IDFT) is given by,2) To verify the above expression, we multiply both sides of the above equation by WNln and sum the result from n = 0 to n=N-1,5.2.1 Definition,11,resulting in,5.
8、2.1 Definition,12,Making use of the identity,Hence,5.2.1 Definition,13,5.2.1 Definition,1) Consider the length-N sequence,Its N-point DFT is given by,4. Example,14,5.2.1 Definition,2) Consider the length-N sequence,Its N-point DFT is given by,4. Example,15,5.2.1 Definition,3) Consider the length-N s
9、equence defined for 0 n N-1,4. Example,Using a trigonometric identity we can write,16,5.2.1 Definition,4. Example,The N-point DFT of xn is thus given by,Making use of the identity,r an integer,17,5.2 .1 Definition,4. Example,18,5.2.2 Matrix Relations,where,The DFT samples defined by:,can be expresse
10、d in matrix forms as:,19,5.2.2 Matrix Relations,DN is the NN DFT matrix given by,20,5.2.2 Matrix Relations,Likewise ,the IDFT can be expressed in matrix forms as:,21,5.2.2 Matrix Relations,is the NN IDFT matrix given by,Note,22,5.3 Relation between DTFT and DFT and their Inverses,5.3.1 Relation with
11、 DTFT,The Fourier transform of the length-N sequence xn.,By uniformly sampling at N equally spaced frequencies,23,5.3.2 Numerical Computation of DTFT,Define a new sequence xen,Eq.(5.38)used function freqz can evaluate the frequency response of xn,24,5.3.3 The DTFT from DFT by interpolation,1.show,25
12、,5.3.3 The DTFT from DFT by interpolation,26,5.3.3 The DTFT from DFT by interpolation,where,27,from Xk by interpolation,5.3.3 The DTFT from DFT by interpolation,Result,The interpolating polynomial satisfies :,28,5.3.4 Sampling the DTFT,1. Consider a sequence xn with a DTFT,2. We sample at N equally
13、spaced points developing the N Frequency samples,3. These N frequency samples can be considered as an N-point DFT Yk whose N-point IDFT is a length-N sequence yn.,29,5.3.4 Sampling the DTFT,Now,From Eq.(3.12),30,5.3.4 Sampling the DTFT,Making use of the above identity in Eq.(5.48),then,yn is obtaine
14、d by adding infinite number of shifted replicas of xn.,Each replica shifted by an integer multiple of N sampling instants,31,5.3.4 Sampling the DTFT,To apply,a) If xn is length-M sequence with MN,thus yn=xn for 0 n N-1.,b) If MN,there is a time-domain aliasing of samples of xn in generating yn, and
15、xn can not be recovered from yn.,32,5.3.4 Sampling the DTFT,Example 5.6,Give 8 equally spaced points,-xn can be recovered from yn,Give 4 equally spaced points,-xn cant be recovered from yn,33,5.4 Operation on Finite-Length Sequences,5.4.1 Circular shift of a sequence,1.Consider length-N sequences de
16、fined for 0nN-1 2.Sample values of such sequences are equal to zero for values of n 0 and nN,34,5.4.1 Circular shift of a sequence,3. If xn is such a sequence, then for any arbitrary integer n0 , the shifted sequence x1n = xn n0 is no longer defined for the range 0nN-1 4. We thus need to define another ty
