Chapter5 Transform-Domain Representation of Discrete-Time Signals

,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Chapter 3 Transform-Domain Representation of Discrete-Time Signals,3.1 Discrete-Time Fourier Transform,Definition,- The discrete-time Fourier transform (,DTFT,),X(e,j,),of a sequence xn is given by,X(e,j,) =,X,re,(e,j,),+ j,X,im,(e,j,),In general,X(e,j,),is a complex function of the real variable,w,and can be written as,3.1 Discrete-Time Fourier Transform,X,re,(e,j,),and,X,im,(e,j,),are, respectively, the real and imaginary parts of,X(e,j,), and are real functions of,w,X(e,j,),can alternately be expressed as,X(e,j,) = |,X(e,j,),|,e,j,(),where,() =,arg,X(e,j,) ,3.1 Discrete-Time Fourier Transform,|,X(e,j,),| is called the,magnitude function,(),is called the,phase function,Both quantities are again real functions of,w,In many applications, the DTFT is called the,Fourier spectrum,Likewise,|,X(e,j,),| and,(),are called the,magnitude,and,phase spectra,3.1 Discrete-Time Fourier Transform,For a real sequence xn,|,X(e,j,),| and,X,re,(e,j,),are even functions of,w, whereas,(),and,X,im,(e,j,),are odd functions of,w,Note,:,X(e,j,) = |,X(e,j,),|e,j,(+2k),=,|,X(e,j,),|,e,j,(),for any integer k,The phase function,q,(,w,) cannot be uniquely specified for any DTFT,3.1 Discrete-Time Fourier Transform,Unless otherwise stated, we shall assume that the phase function,q,(,w,) is restricted to the following range of values:,-,q,(,w,), ,called the,principal value,3.1 Discrete-Time Fourier Transform,The,DTFTs,of some sequences exhibit discontinuities of 2,p,in their phase responses,An alternate type of phase function that is a continuous function of,w,is often used,It is derived from the original phase function by removing the discontinuities of 2,p,3.1 Discrete-Time Fourier Transform,Example,- The DTFT of the unit sample sequence,d,n, is given by,Example,- Consider the causal sequence,3.1 Discrete-Time Fourier Transform,Its DTFT is given by,as,3.1 Discrete-Time Fourier Transform,The magnitude and phase of the DTFT,X(e,j,),= 1/(1 0.5e,-j,),are shown below,3.1 Discrete-Time Fourier Transform,The DTFT,X(e,j,),of a sequence xn is a continuous function of,w,It is also a periodic function of,w,with a period 2,p,:,3.1 Discrete-Time Fourier Transform,Therefore,As a result, the Fourier coefficients xn can be computed from,X(e,j,),using the Fourier integral,represents the Fourier series representation of the periodic function,3.1 Discrete-Time Fourier Transform,Inverse discrete-time Fourier transform:,Proof:,3.1 Discrete-Time Fourier Transform,The order of integration and summation can be interchanged if the summation inside the brackets converges uniformly, i.e.,X(e,j,),exists,Then,3.1 Discrete-Time Fourier Transform,Now,Hence,3.1 Discrete-Time Fourier Transform,Convergence Condition - An infinite series of the form,may or may not converge,Let,3.1 Discrete-Time Fourier Transform,Then for uniform convergence of,X(e,j,),Now, if xn is an absolutely,summable,sequence, i.e., if,3.1 Discrete-Time Fourier Transform,Then,Thus, the absolute,summability,of xn is a,sufficient condition,for the existence of the DTFT,X(e,j,),for all values of,w,3.1 Discrete-Time Fourier Transform,Example,- The sequence xn =,n,n,for |,| 1,is absolutely,summable,as,and its DTFT,X(e,j,),therefore converges to 1/(1-,e,-j,),uniformly,3.1 Discrete-Time Fourier Transform,Since,However, a finite-energy sequence is not necessarily absolutely,summable,an absolutely,summable,sequence has always a finite energy,3.1 Discrete-Time Fourier Transform,Example,- The sequence,E,But, xn is not absolutely,summable,has a finite energy equal to,3.1 Discrete-Time Fourier Transform,A,Dirac,delta function,d,(,w,) is a function of,w,with infinite height, zero width, and unit area,w,It is the limiting form of a unit area pulse function p,(),as,D,goes to zero satisfying,3.1 Discrete-Time Fourier Transform,Example,- Consider the complex exponential sequence,where,d,(,w,) is an impulse function of,w,and,Its DTFT is given by,3.1 Discrete-Time Fourier Transform,is a periodic function of,w,with a period 2,p,and is called a,periodic impulse train,or,impulse train,To verify that,X(e,j,),given above is indeed the DTFT of xn=e,j,0,n,we compute the inverse DTFT of,X(e,j,),The function,3.1 Discrete-Time Fourier Transform,Thus,where we have used the sampling property of the impulse function,(),Commonly Used DTFT Pairs,Sequence DTFT,3.2,DTFT Properties,There are a number of important properties of the DTFT that are useful in signal processing applications,These are listed here without proof,Their proofs are quite straightforward,We illustrate the applications of some of the DTFT properties,3.2,DTFT Properties,Type of Property Sequence DTFT,Parsevals,relation,Modulation gnhn,Convolution gn*hn,G(e,j,)H,(e,j,),Differentiation,n,gn,jd,G(e,j,)/d,Frequency-shifting,e,-j,0,n,gn,G(e,j,(,- ,0,),),Time-shifting gn-n,0,e,-j,n,0,G(e,j,),Linearity,agn+bhn,a,G(e,j,)+bH,(e,j,),hn,H(e,j,),gn,G(e,j,),3.2,DTFT Properties,gn,G(e,j,),Use the definition of,G(e,j,)and differentiate both sides, we obtain,The right-hand side of this equation is the Fourier transform of ,jngn,. Therefore, multiplying both sides by j, we see,ngnjdG(ej,)/d,3.2,DTFT Properties,Example,- Determine the DTFT,Y(e,j,),of yn=(n+1),n,n, |1,Let xn=,n,n, |1,We can therefore write,yn=,nxn, + xn,The DTFT of xn is given by,3.2,DTFT Properties,Using the differentiation property of the DTFT, we observe that the DTFT of,nxn,is given by,Next using the linearity property of the DTFT we arrive at,3.2,DTFT Properties,Example,- Determine the DTFT,V(e,j,),of the sequence,v,n, defined by,d,0,vn+d,1,vn-1 = p,0,n + p,1,n-1,The DTFT of,n,is 1,Using the time-shifting property of the DTFT we observe that the DTFT of,n-1,is e,-j,and the DTFT of,v,n-1 is,e,-j,V(e,j,),3.2,DTFT Properties,Using the linearity property we then obtain the frequency-domain representation of,d,0,vn+d,1,vn-1 = p,0,n + p,1,n-1,as d,0,V(e,j,)+,d,1,e,-j,V(e,j,) =,p,0,+ p,1,e,-j,Solving the above equation we get,V(e,j,) =(,p,0,+ p,1,e,-j,)/(,d,0,+d,1,e,-j,),3.2,DTFT Properties,The total energy of a finite-energy sequence,g,n, is given by,E,E,From,Parsevals,relation we observe that,3.2,DTFT Properties,is called the,energy density spectrum,The area under this curve in the range -,divided by 2,p,is the energy of the sequence,The quantity,DTFT Computation Using MATLAB,The function,freqz,can be used to compute the values of the DTFT of a sequence, described as a rational function in the form of,at a prescribed set of discrete frequency points,=,DTFT Computation Using MATLAB,For example, the statement,H =,freqz,(num, den, w),returns the frequency response values as a vector H of a DTFT defined in terms of the vectors,num,and,den,containing the coefficients p,i, and ,d,i, respectively at a prescribed set of frequencies between 0 and 2,p,given by the vector w,DTFT Computation Using MATLAB,There are several other forms of the function,freqz,The,Program 3_1,(p.128) in the text can be used to compute the values of the DTFT of a real sequence,It computes the real and imaginary parts, and the magnitude and phase of the DTFT,DTFT Computation Using MATLAB,Example,- Plots of the real and imaginary parts, and the magnitude and phase of the DTFT,are shown on the next slide,DTFT Computation Using MATLAB,DTFT Computation Using MATLAB,Note,: The phase spectrum displays a discontinuity of 2,p,at,w,= 0.72,This discontinuity can be removed using the function,unwrap,as indicated below,Linear Convolution Using DTFT,An important property of the DTFT is given by the convolution theorem,It states that if yn = xn*hn, then the DTFT,Y(e,j,),of yn is given by,Y(e,j,) =,X(e,j,),H(e,j,),An implication of this result is that the linear convolution yn of the sequences xn and hn can be performed as follows:,Linear Convolution Using DTFT,1) Compute the,DTFTs,X(e,j,),and,H(e,j,),of the sequences xn and hn, respectively,2) Form the DTFT,Y(e,j,) =,X(e,j,),H(e,j,),3) Compute the IDTFT yn of,Y(e,j,),xn,hn,yn,DTFT,DTFT,IDTFT,X(e,j,),H(e,j,),Y(e,j,),3.3 Discrete Fourier Transform (DFT),DTFT is the Fourier Transform of discrete-time sequence. It is discrete in time domain and its spectrum is periodical, but continue which cannot be processed by computer which could only process digital signals in both sides, that means the signals in both sides must be both discrete and periodical.,3.3 Discrete Fourier Transform (DFT),Time domain Frequency domain,Continue,aperiodical,FT,Continue,aperiodical,Periodical,FST,discrete spectrum,Discrete,DTFT,periodical spectrum,Discrete periodical,DFT,periodical discrete,Typical DFT Pair,T,(t),0,0,(),T,2T,-T,-2T,T,(t),t,0,0,2,0,-,0,-2,0,0,0,(,),0,0,= 2/T,In DFT, the signals in both sides are discrete, so it is the only transform pair which can be processed by computer.,The signals in both sides are periodical, so the processing could be in one period, which is important because (1) the number of calculation is limited, which is necessary for computer; (2) all of the signal information could be kept in one period, which is necessary for accurate processing.,Make a signal discrete and periodical,The engineering signals are often continue and,aperiodical,. If we want to process the signals with DFT, we have to make the signals discrete and periodical,.,Sampling to make the signal discrete,Make the signal periodical:,If xn is a limited length N-point sequence, see it as one period of a periodical signal that means extend it to a periodical,If xn is an infinite length sequence, cut-off its tail to make a N-point sequence, then do the periodic extending. The tail cutting-off will introduce distortion. We must develop truncation algorithm to reduce the error, which is windowing.,Make a signal discrete and periodical,x(t),X(j,),P(,j,),0,0,=2/T,s,xnT,X(j,),q(,t,),T,FT,DFT,DTFT,Q(,j,),0,0,=,2/T,Q(,j,),0,0,=,2/T,P(t),T,s,3.3 Discrete Fourier Transform (DFT),Definition,- The simplest relation between a length-N sequence xn, defined for 0,n,N-1, and its DTFT,X(e,j,),is obtained by uniformly sampling,X(e,j,),on the,w,-axis between 0,2,at,k,=2k/N, 0,k,N-1,From the definition of the DTFT have,3.3 Discrete Fourier Transform (DFT),Note,: Xk is also a length-N sequence in the frequency domain,The sequence Xk is called the,discrete Fourier transform (DFT),of the sequence xn,Using the notation W,N,=e,-j2,/N,the DFT is usually expressed as:,3.3 Discrete Fourier Transform (DFT),The,inverse discrete Fourier transform (IDFT),is given by,To verify the above expression we multiply both sides of the above equation by,W,N,ln,and sum the result from n = 0 to n=N-1,3.3 Discrete Fourier Transform (DFT),resulting in,3.3 Discrete Fourier Transform (DFT),Making use of the identity,r,an integer,Hence,3.3 Discrete Fourier Transform (DFT),Example,- Consider the length-,N,sequence,Its,N,-point DFT is given by,3.3 Discrete Fourier Transform (DFT),Example,- Consider the length-,N,sequence,Its,N,-point DFT is given by,3.3 Discrete Fourier Transform (DFT),Example,- Consider the length-,N,sequence defined for 0,n N-1,Using a trigonometric identity we can write,3.3 Discrete Fourier Transform (DFT),The,N,-point DFT of,g,n, is thus given by,3.3 Discrete Fourier Transform (DFT),Making use of the identity,r,an integer,we get,DFT Computation Using MATLAB,The functions to compute the DFT and the IDFT are,FFT,and,IFFT,These functions make use of FFT algorithms which are computationally highly efficient compared to the direct computation,Programs 3_2,(p.134) and,3_4,(p.136) illustrate the use of these functions,DFT Computation Using MATLAB,Example,-,Program 3_4,can be used to compute the DFT and the DTFT of the sequence,indicates DFT samples,as shown below,3.4 DFT Properties,Like the DTFT, the DFT also satisfies a number of properties that are useful in signal processing applications,Some of these properties are essentially identical to those of the DTFT, while some others are somewhat different,A summary of the DFT properties are given in tables in the following slides,3.4,DFT Properties,Type of Property length-N sequence N-point DFT,Parsevals,relation,Modulation gnhn,Gk,H,k,Circular Convolution,Duality G,n,Ng-k,N,Frequency-shifting W,N,-,k,n,0,gn,G,k-k,0,N,Circular Time-shifting gn-n,0,N, W,N,k,n,0,Gk,Linearity,agn+bhn,a,Gk,+bH,k,hn Hk,gn Gk,3.5,Circular Shift of a Sequence,This property is analogous to the time-shifting property of the DTFT , but with a subtle difference,Consider length-,N,sequences defined for,0,n,N-1,Sample values of such sequences are equal to zero for values of,n,0 (right circular shift), the above equation implies,3.5,Circular Shift of a Sequence,Illustration of the concept of a circular shift,3.5,Circular Shift of a Sequence,As can be seen from the previous figure, a right circular shift by n,0,is equivalent to a left circular shift by N-n,0,sample periods,A circular shift by an integer number greater than,N,is equivalent to a circular shift by,n,0,N,3.5,Circular Shift of a Sequence,xn-1,xn,x,N,n=,4,=0,3.6,Circular Convolution,This operation is analogous to linear convolution, but with a subtle difference,Consider two length-N sequences, gn and hn, respectively,Their linear convolution results in a length-(2N-1) sequence,y,L,n, given by,3.6,Circular Convolution,In computing,y,L,n, we have assumed that both length-,N,sequences have been zero-padded to extend their lengths to 2N-1,The longer form of,y,L,n, results from the time-reversal of the sequence,h,n, and its linear shift to the right,The first nonzero value of,y,L,n, is,y,L,n,=g0h0, and the last nonzero value is,y,L,2N-2=gN-1hN-1,3.6,Circular Convolution,To develop a convolution-like operation resulting in a length-,N,sequence,y,C,n, we need to define a circular time-reversal, and then apply a circular time-shift,Resulting operation, called a,circular convolution, is defined by,3.6,Circular Convolution,Since the operation defined involves two length-N sequences, it is often referred to as an N-point circular convolution, denoted as,N,N,gn hn = hn gn,The circular convolution is commutative, i.e.,N,yn = gn hn,3.6,Circular Convolution,Example,- Determine the 4-point circular convolution of the two length-4 sequences:,n,n,as sketched below,3.6,Circular Convolution,The result is a length-4 sequence,y,C,n, given by,4,From the above we observe,3.6,Circular Convolution,Likewise,3.6,Circular Convolution,The circular convolution can also be computed using a DFT-based approach,3.6,Circular Convolution,Example,- Consider the two length-4 sequences repeated below for convenience:,n,n,The 4-point DFT Gk of gn is given by,3.6,Circular Convolution,Therefore,Likewise,3.6,Circular Convolution,Hence,Homework,Read textbook from p.117 to 155,Problems,3.1, 3.4, 3.6, 3.8, 3.13, 3.18, 3.22, 3.25, 3.50, 3.51, 3.56, 3.64, 3.65, 3.74,M3.2, M3.8, M3.9,