∑对∑pN∑m-n k=0 3.39(a) yIn]=agIn]+Bh(n]. Therefore N-1 Yk=∑yW=a2w+p∑w时=0出k+BHk (b)x[n]=gk< n-no >NI. Therefore X[k]= >el< n-no >NJWN ∑N+n-nw+∑gn- noWN N-n-1 =∑gnw+n。N+∑gnwN)=wk∑gnw=wRG以k ()ml= WNgIn]. Hence U[k]=∑w=∑nwN-0 n=0 ∑ gn],ifk≥k ∑ (N+k-ko) gn], if k<k Thus, UIk]=GiN+k-kol, if k<ko, = Gk< k-ko >NI d)hnl=Gn. Therefore, Hlk]=∑hnw=∑Gnw=∑2 g[] wkr ∑gr∑w(k+rn The second sum is non-zero only if k =r=0 or else if r=n-k and k*0. Hence, Nglo], if HIkJ-NgIN-k], if k>0=Ngk<-k>NI e)unl=∑smH<nm> 1. Therefore,k=∑∑s >、W ∑gm∑h<nm>Nw=∑ glm]HIkM=HkGk
57 = ˜[ ] ˜ [ ] ( )/ x r N Ykej kn r N k N r N 1 2 0 1 0 1 π − = − = − ∑ ∑ = xr yn r ˜[ ]˜[ ] r N − = − ∑ 0 1 . 3.39 (a) yn gn hn [] [] [] = + α β . Therefore Yk yn W gnW hnW Gk Hk N nk n N N nk n N N nk n N [] [] [] [] [] [] = = + =+ = − = − = − ∑ ∑∑ 0 1 0 1 0 1 α β αβ (b) xn g n n [] [ ] =<− > 0 N . Therefore X[k] = g nn WN N nk n N [ ] <− > = − ∑ 0 0 1 = gN n n WN nk n n [ ] + − = − ∑ 0 0 1 0 + gn n WN nk n n N [ ] − = − ∑ 0 1 0 = gnW gnW N n n Nk nNn N N nn k n N n [] [] ( ) () + − = − − + = − − ∑ ∑ + 0 0 0 1 0 0 1 = W gnW N n k N nk n N 0 0 1 [ ] = − ∑ = W Gk N n k0 [ ]. (c) un W gn N k n [] [] = − 0 . Hence U k u n W g n W N nk n N N kk n n N [] [] [] ( ) = = = − − = − ∑ ∑ 0 1 0 1 0 = ≥ < − = − + − = − ∑ ∑ W g n if k k W g n if k k N kk n n N N Nkk n n N ( ) ( ) [ ], , [ ], . 0 0 0 1 0 0 1 0 Thus, U k G k k if k k G N k k if k k [ ] [ ], , [ ], , = − ≥ +− < 0 0 0 0 = G kk [ ] <− > 0 N . (d) h[n] = G[n]. Therefore, Hk hnW GnW N nk n N N nk n N [] [] [] = = = − = − ∑ ∑ 0 1 0 1 = = − = − ∑ ∑grW Wnr kr r N n N [ ] 0 1 0 1 = gr W k rn n N r N [ ] ( ) + = − = − ∑ ∑ 0 1 0 1 . The second sum is non-zero only if k = r = 0 or else if r = N – k and k ≠ 0. Hence, H[k] = Ng if k Ng N k if k Ng k N [ ], , [ ], , [ ] 0 0 0 = − > = <− > . (e) un gmh n m N m N [] [ ][ ] = <− > = − ∑ 0 1 . Therefore, U k g m h n m W N m N n N N nk [] [ ][ ] = <− > = − = − ∑ ∑ 0 1 0 1 = gm h n m WN n N m N N nk [] [ ] <− > = − = − ∑ ∑ 0 1 0 1 = = − ∑gmHkW m N N mk [ ][] 0 1 = H[k]G[k]
(f)vIn=g[n]h(n]. Therefore, v[k]=>gIn]h[n]wnk I NI InGr] Wnw n=0r=0 ()xnN∑xw. Thus x+N∑x+kw. Therefore N-1 -1N-1 ∑n=∑∑ⅪwΣxwa∑∑Ⅺx∑w= 1=0 r=01=0 Since the inner sum is non-zero only if 1=;wegt∑xmn=∑xk 3.40Xk]=∑xnw (a)x*[]=2x*[n]w-nk. Replacing k by N-k on both sides we obtain xN-k=∑xnwm=∑xnwk, Thus台xN-k=x"kk>N (b)X(k]=2xInjw-nk. Replacing n by N-n in the summation we get n=0 xk=∑xN-nNN-=∑x"Nnw吐 n=0 Thus xIN-n=x k<-n>NoX k (c)Re([n=(x[n]+x[n]. Now taking DFT of both sides and using results of part(a) we get Re(x叫]分(X队k]+x*k-k> (d)j Im(x[n])=-(x(n]-x*(n]) this imples j Im(x[n])+-(X(k]-X K<-k >NII (e)XpcsIn]=xIn]+x[<-n >NI/ USing linearity and results of part(b) we get Xpcs[n] +3 xIk]+Xlk]=Re[XIk] (xpcaln=lxn]-x(<-n >NIl. Again using results of part(b)and linearity we aln] +(XIk]-X [k])=j Im (X(k])
58 (f) v[n] = g[n]h[n]. Therefore, V k g n h n W n N N nk [] [][] = = − ∑ 0 1 = = − = − = − ∑ ∑ 1 0 1 0 1 N hnGr W Wnk nr r N n N [ ] [] 1 1 0 1 0 1 0 1 N Gr hnW N GrH k r k rn n N r N N r N [] [ ] [] [ ] ( ) − = − = − = − ∑ ∑ ∑ = < −> . (g) x n N XkW nk k N [] [] . = 1 − = − ∑ 0 1 Thus x n N X kWnk k N *[ ] *[ ] . = 1 = − ∑ 0 1 Therefore, x n N XrW X W n N nr r N n N n N [ ] [ ] *[ ] 2 0 1 2 0 1 0 1 0 1 1 = − − = − = − = − ∑ ∑ = ∑ ∑ l l l = − = − = − = − ∑ ∑ ∑ 1 2 0 1 0 1 0 1 N Xr X Wn r n N N r N [] [] * () l l l . Since the inner sum is non-zero only if l= r, we get x n N X k n N k N [] [] . 2 0 1 2 0 1 1 = − = − ∑ ∑ = 3.40 X[k] = xnWnk n N [ ] = − ∑ 0 1 . (a) X*[k] = x n W nk n N *[ ] . − = − ∑ 0 1 Replacing k by N – k on both sides we obtain X*[N – k] = x n W x n W nN k n N n N nk *[ ] *[ ] . − − ( ) = − = − ∑ ∑= 0 1 0 1 Thus x*[n] ⇔ X*[N – k] = X*[< –k >N]. (b) X*[k] = x n W nk n N *[ ] . − = − ∑ 0 1 Replacing n by N – n in the summation we get X*[k] = x N n W x N n W N nk n N nk n N *[ ] *[ ] . ( ) − =− − − = − = − ∑ ∑ 0 1 0 1 Thus x*[N – n] = x*[< –n >N] ⇔ X*[k]. (c) Re{x[n]} = 1 2 {x[n] + x*[n]}. Now taking DFT of both sides and using results of part (a) we get Re{x[n]} ⇔ 1 2 {X[k] + X*[< –k >N]}. (d) j Im{x[n]} = 1 2 {x[n] – x*[n]} this imples j Im{x[n]}⇔ 1 2 {X[k] – X*[< –k >N]}. (e) xpcs[n] = 1 2 {x[n] + x*[<–n >N]} Using linearity and results of part (b) we get xpcs[n] ⇔ 1 2 {X[k] + X*[k]} = Re{X[k]}. (f) xpca[n] = 1 2 {x[n] – x*[< –n >N]}. Again using results of part (b) and linearity we get xpca[n] ⇔ 1 2 {X[k] –X*[k]} = j Im {X[k]}
341x=R(Xk]+jm(xk]:∑刈-P2 n=0 DET (a)xpe[n]=3xn]+xk<->N]. From Table 3.6, x*K<-n>NeX*(k]. Since xIn) real, xk<-n> 3。Cb)xm=m-x<-nx小 As a result,xk]=x]-x+=jmxk] Since for a real sequence, x[n]=x"[n], taking DFT of both sides we get X[k] X[<-kN]. This implies, Re(xkk]+j Im(x(k])= Re(x(<- -j Im(X<-kNJ Comparing real and imaginary parts we get Relxik)= relx[<- kN and Im(xIk]=-ImXl<-kNJ Also x(k]=(Re(X(kJ)2+(Im(X(kJ) RX+-x22-xx and arg X[k]=tan Im{ⅹ[k]} -Im(X(<-k>NII Reek ReX<-k>NIl -argX(<-k>INI 3.43(a)x1[<-n>8]=11 10001 1=x1[n]. Thus, xi[n] is a periodic even sequence. nd hence it has a real-valued 8-point DFT. (b)x2(<-n>81=[1 -1-10000 1]. Thus, x2[n] is neither a periodic even or a periodic odd sequence. Hence, its 8-point DFT is a complex sequence. (c)x3(<-n>8]=0-1 -1000 11=-x3In]. Thus, x3[] is a periodic odd sequence, and hence it has an imaginary-valued 8-point DFT. (d)x4[<-n>81=[0 1 1000 11=x4[n]. Thus, X4(n] is a periodic even sequence, nd hence it has a real-valued 8-point DFT. 3.44(a)Now, XIN/2)=>[]WNN/2=>G1)x[n]. Hence if x[n]=xIN-1-n]and N is n=0 even,then 2(1)"x[n]=0 or X[N/2]=0 n=0 (b)X[O]=2x[n] so if x[n]=-xIN-1-nl, then X[0]=0 n=0
59 3.41 X[k] = Re{X[k]} + j Im{X[k]} = x n e j kn N n N [] . – / 2 0 1 π = − ∑ (a) x n xn x n pe N [ ] [ ] [ ]. = + <− > { } 1 2 From Table 3.6, x n X k N DFT *[ ] *[ ]. <− > ⇔ Since x[n] is real, x n x n X k N N DFT [ ] *[ ] *[ ]. <− > = <− > ⇔ Thus, X k Xk X k Xk pe[ ] [ ] *[ ] Re{ [ ]}. = + { } = 1 2 (b) x n xn x n po N [ ] [ ] [ ]. = − <− > { } 1 2 As a result, X k Xk X k j Xk po[ ] [ ] *[ ] Im{ [ ]}. = − { } = 1 2 3.42 Since for a real sequence, x[n] = x*[n], taking DFT of both sides we get X[k] = X*[<– k>N]. This implies, Re{X[k]} + j Im{X[k]} = Re{X[<– k>N} – j Im{X[<– k>N}. Comparing real and imaginary parts we get Re{X[k]} = Re{X[<– k>N} and Im{X[k]} = – Im{X[<– k>N}. Also Xk Xk Xk [ ] Re{ [ ]} Im{ [ ]} = ( ) + ( ) 2 2 = <> (Re{ [ – ]} – Im{ [ – ]} Xk Xk N N ) + <> ( ) 2 2 =< > X k N [– ] and arg{ [ ]} tan Im{ [ ]} Re{ [ ]} tan – Im{ [ ]} Re{ [ ]} X k X k X k X k X k N N = = <− > <− > − − 1 1 =− <− > arg{ [ ] } X k N . 3.43 (a) x n xn 18 1 [ ] [ ]. <− > = [ ] 1110 0 011 = Thus, x n 1[ ] is a periodic even sequence, and hence it has a real-valued 8-point DFT. (b) xn 2 8 [ ] .. <− > = − − [ ] 1 1 100001 Thus, x n 2[ ] is neither a periodic even or a periodic odd sequence. Hence, its 8-point DFT is a complex sequence. (c) x n xn 38 3 [ ] [ ]. <− > = − − [ ] 0 1 10 0 011 = − Thus, x n 3[ ] is a periodic odd sequence, and hence it has an imaginary-valued 8-point DFT. (d) x n xn 48 4 [ ] [ ]. <− > = [ ] 0110 0 011 = Thus, x n 4[ ] is a periodic even sequence, and hence it has a real-valued 8-point DFT. 3.44 (a) Now, X[N/2] = xnW xn N nN n N n n N [] ( ) [] / 2 0 1 0 1 1 = − = − ∑ ∑= − . Hence if x[n] = x[N – 1 – n] and N is even, then ( ) [] − = = − ∑ 1 0 0 1 n n N x n or X[N/2] = 0. (b) X[0] = x n n N [ ] = − ∑ 0 1 so if x[n] = – x[N – 1 – n], then X[0] = 0
(c)x2=∑xnW ∑ ∑ xn/w2n1 n=N/2 =∑n2+∑xn+2N2=∑(m+xm+w2n n=0 Hence if xn=-xn+=-xn+m, then X(21]=0, for 1=0, 1,...., M-1 N-1 3.45X2m=∑x叫wm=∑xwm+∑xmwm N ∑x叫wm+∑ N2m(n+亏) ∑x叫Wm+∑xn+ WNWN n=0 ∑(m+xm+1wm=0.0≤m≤-1. This implies x[n]+xm+]=0 n=0 2 3.46(a) Using the circular time-shifting property of the dFt given in Table 3.5 we observe DFTx(<n-mI >N=WN IX[k] and DFTxK<n-m2 >N=WN 2X(k].Hence, WIk]=DFT(x[nl)=awkmI XIk]+Bwkm2x[k]=(awk I Bwkm2 )x(k]. A proof of the circular time-shifting property is given in Problem 3, 39. (b)gln]=(xn]+(-1)"x(n [n]+ WN x[n] Using the circular frequency-shifting property of the DFT given in Table 3.5, we get G[k]=DFT(gIn]=XL (c) Using the circular convolution property of the dFt given in Table 3. 5 we get Y[k]=DFTy(n])=X[k]. XIk]=X[k]. A proof of the circular convolution property is given in Problem 3.39 3.47(a) DFt[n WN2[k]=-XIk]Hence u[k]=DFTu[n))=DFTx[n]+x[n-1=X[k]-X[k]=0 b)Vk]=DFvm}=DFxn]-xn-}=xk1+Xk]=2Xk
60 (c ) X xn W xnW xnW n N n n n n N N n N [ ] [] [] [] / 2 0 1 2 0 1 2 2 1 2 2 l ll l = =+ = − = − = − ∑∑∑ = + + = ++ = − = − = − ∑∑ ∑ xnW xn W xn xn W n n N n n n N n NN N [ ] [ ] ( [ ] [ ]) 0 1 2 2 0 1 2 0 1 2 2 22 2 ll l. Hence if x[n] = – x[n + N 2 ] = – x[n+M], then X[2 l] = 0, for l = 0, 1, . . . . , M – 1. 3.45 X m xnW xnW xnW N mn n N N mn n N N mn n N N [ ] [] [] [] 2 2 0 1 2 0 2 1 2 2 1 == + = − = − = − ∑∑∑ = ++ = ++ = − + = − = − = − ∑∑ ∑∑ xnW xn N W xnW xn N N mn W W n N N m n N n N N mn n N N mn n N N mN [] [ ] [] [ ] ( ) 2 0 2 1 2 2 0 2 1 2 0 2 1 2 0 2 1 2 2 = ++ = ≤≤ − = − ∑ xn xn N W m N N mn n N [] [ ] , . 2 0 0 2 1 2 0 2 1 This implies xn xn N [] [ ] . + += 2 0 3.46 (a) Using the circular time-shifting property of the DFT given in Table 3.5 we observe DFT x n m W X k N N km { } [ [] <− >1 = 1 and DFT x n m W X k N N km { } [ [ ]. <− > 2 = 2 Hence, W k DFT x n W X k W X k W W X k N km N km N km N km [ ] [ ] [ ] [ ] [ ]. = { } = + =+ α β αβ ( ) 1 2 12 A proof of the circular time-shifting property is given in Problem 3,39. (b) gn xn xn xn W xn n N N n [ ] [ ] ( ) [ ] [ ] [ ], = +− ( ) = + − 1 2 1 1 2 2 Using the circular frequency-shifting property of the DFT given in Table 3.5, we get G k DFT g n X k X k N [ ] [ ] [ ] [ ]. = { } = + <− >N 1 2 2 (c) Using the circular convolution property of the DFT given in Table 3.5 we get Y k DFT y n X k X k X k [ ] [ ] [ ] [ ] [ ]. = { } =⋅= 2 A proof of the circular convolution property is given in Problem 3,39. 3.47 (a) DFT x n N W Xk Xk N k N [ ] [ ] [ ]. − = =− 2 2 Hence, u k DFT u n DFT x n x n N [] [] [] [ ] [] [] . = { } = +− Xk Xk =−= 2 0 (b) V k DFT v n DFT x n x n N [ ] [ ] [ ] [ ] [ ] [ ] [ ]. = { } = −− Xk Xk Xk =+= 2 2
(c)yIn]=(D"x[n]=WN x[n]. Hence, Y[k]=DFT y(n]=DFT WE X[n1=X(<k-N using the circular frequency-shifting property of the dFt given in Table 3.5 3.48(a) From the circular frequency-shifting property of the dFT given in Table 3.5 IDFT XI<k-mINJ=WN Ix[n] and IDFT XI<k-m2 >NI=WN 2 x[n]. Hence, w[]=IDFT(W[])=IDFToXI<k-m1>N+BXI<k-m2 >NI =aWxa+BwN=(w吗+Bwm (b)G[k]=X[k]+(-1)X[k]= X[k]+WN 2 X[k] Using the circular time-shifting property of the DFT given in Table 3.5, we get gIn]=IDFT(G[k]=5 x[n]+xk<n->N (c) Using the modulation property of the dft given in Table 3. 5 we get y[n]=IDFT(Y[]=Nx[n] x[n]=NX-In 349(a)X2m=∑xwm=∑xwm+∑x叫W ∑x叫Wm+∑Ⅺn+W2=∑xwm+∑x+wmwN n=0 ∑xm]+xn+1wm=∑(对-xn)wm=0.0≤msN n=0 N (b)X41=∑xmlw=∑xmw+∑xmw+∑xnw+∑xnlw n=0 n=0 1(n+ >IxnJwNn+x[n+IWN 4+[n+JWN + xln 2x[n]+x[n+ jWN+x[n+SIWEN+x[n+WNN N n=0 =∑(x]-xn+x叫-xm)w=0asw=w=WN=1
61 (c) yn xn W xn n N N n [ ] ( ) [ ] [ ]. =− = 1 2 Hence, Y k DFT y n DFT W x n X k N N N n [] [] [] [ ] = { } = N = <− > 2 2 using the circular frequency-shifting property of the DFT given in Table 3.5. 3.48 (a) From the circular frequency-shifting property of the DFT given in Table 3.5, IDFT X k m W x n N N m n { } [ ] [] <− > = − 1 1 and IDFT X k m W x n N N m n { } [ ] [ ]. <− > = − 2 2 Hence, w n IDFT W k IDFT X k m X k m [] [] [ [ = { } = <− >+ <− > { } α β 1 2 N N = + =+ ( ) − − −− α β αβ W xn W xn W W xn N m n N m n N m n N 1 2 12 m n [ ] [ ] [ ]. (b) Gk Xk Xk Xk W Xk k N N k [ ] [ ] ( ) [ ] [ ] [ ]. = +− ( ) = + − 1 2 1 1 2 2 Using the circular time-shifting property of the DFT given in Table 3.5, we get g n IDFT G k x n x n N [ ] [ ] [ ] [ ]. = { } = +<− >N 1 2 2 (c) Using the modulation property of the DFT given in Table 3.5 we get y n IDFT Y k N x n x n N x n [ ] [ ] [ ] [ ] [ ]. = { } =⋅ ⋅ =⋅ 2 3.49 (a) X m xnW xnW xnW N mn n N N mn n N N mn n N N [ ] [] [] [] 2 2 0 1 2 0 2 1 2 2 1 == + = − = − = − ∑∑∑ = ++ = ++ = − + = − = − = − ∑∑ ∑∑ xnW xn N W xnW xn N N mn W W n N N m n N n N N mn n N N mn n N N mN [] [ ] [] [ ] ( ) 2 0 2 1 2 2 0 2 1 2 0 2 1 2 0 2 1 2 2 = ++ = − ( ) = ≤≤ − = − = − ∑ ∑ xn xn N W xn xn W m N N mn n N N mn n N [] [ ] [] [] , . 2 0 0 2 1 2 0 2 1 2 0 2 1 (b) X xnW xnW xnW xnW xnW N n n N N n n N N n n N N N n n N N N n n N N [ ] [] [] [] [] [] 4 4 0 1 4 0 4 1 4 4 2 1 4 2 3 4 1 4 3 4 1 l ll l l l == + + + = − = − = − = − = − ∑∑∑ ∑ ∑ = ++ ++ ++ ++ + = − ∑ xnW xn N W xn N W xn N N n N W n N N n N N n N n N [] [ ] [ ] [ ] () () ( ) 4 4 4 4 2 4 3 4 0 4 1 4 2 3 4 l ll l = ++ ++ ++ = − ∑ xn xn N W xn N W xn N N N N N W W N N n N N n [] [ ] [ ] [ ] 4 2 3 4 2 3 0 4 1 l l ll4 = −+− ( ) = = − ∑ xn xn xn xn W n N N n [] [] [] [] 0 4 1 4 0 l as WW W N N N N N lllN === 2 3 1