说明：这两个小题的数学证明过程都不会，欢迎博友赐教。

        直接上代码：

%% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++%%            Output Info about this m-filefprintf('\n***********************************************************\n');fprintf('        
     
       Problem 5.14 \n\n');banner();%% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++% ------------------------------------------------------------------------%              1  x(n) = [1, 2, 3, 1,2,3, 1, 2, 3, 1, 2, 3]   N=12  v=3%                   x(n) = x(n+v)  N=4v                                  % ------------------------------------------------------------------------ nn1 = [0:11];xx1 = [1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3];NN1 = length(xx1);                                % length is 12%m = mod_1(nn1, NN1);%x = [xx1 zeros(1, 0)];                        % padding zeros%n = [nn1 max(nn1)+1:max(nn1)+6];x = xx1;n = nn1;figure('NumberTitle', 'off', 'Name', 'P5.14.1 x(n)')set(gcf,'Color','white'); subplot(2,1,1); stem(nn1, xx1);xlabel('n'); ylabel('x(n)');title('x(n) ori sequence');  grid on;subplot(2,1,2); stem(n, x);xlabel('n'); ylabel('x(n)');title('x(n) padding 0 zeros');  grid on;%% =============================================================================%%                 DTFT X(w) of xn sequence, w=[0:2pi], %% =============================================================================MM = 500;[Xw_DTFT, w] = dtft1(x, n, MM); magXw_DTFT = abs(Xw_DTFT);   angXw_DTFT = angle(Xw_DTFT)/pi; realXw_DTFT = real(Xw_DTFT); imagXw_DTFT = imag(Xw_DTFT);%% --------------------------------------------------------------%%        START X_DTFT's  mag ang real imag%% --------------------------------------------------------------figure('NumberTitle', 'off', 'Name', 'P5.14.1 X(w) DTFT of x(n)');set(gcf,'Color','white'); subplot(2,2,1); plot(w/pi,magXw_DTFT); grid on;  % axis([-2,2,0,15]); title('Magnitude Part');xlabel('frequency in \pi units'); ylabel('Magnitude  |X\_DTFT|'); subplot(2,2,3); plot(w/pi, angXw_DTFT); grid on;  % axis([-2,2,-1,1]);title('Angle Part');xlabel('frequency in \pi units'); ylabel('Rad \pi'); %axis([-200,200,0,2]);subplot('2,2,2'); plot(w/pi, realXw_DTFT); grid on;title('Real Part');xlabel('frequency in \pi units'); ylabel('Real');subplot('2,2,4'); plot(w/pi, imagXw_DTFT); grid on;title('Imaginary Part');xlabel('frequency in \pi units'); ylabel('Imaginary');%% --------------------------------------------------------------%%        END X_DTFT's  mag ang real imag%% --------------------------------------------------------------%% ------------------------------------------------------------------%%                 DFT(k) of xn sequence, k=[0:N-1]%%                      w=2pi*k/N       k=Nw/(2pi)%% ------------------------------------------------------------------N1 = length(x);k1 = [0 : N1-1];%k2 = [-N : N-1];%k3 = [-N/2 : N/2];Xk_DFT = dft(x, N1);                                         % DFT    magXk_DFT = abs( [ Xk_DFT ] );                          % DFT magnitude    angXk_DFT = angle( [Xk_DFT] )/pi;                       % DFT angle   realXk_DFT = real(Xk_DFT); imagXk_DFT = imag(Xk_DFT);figure('NumberTitle', 'off', 'Name', 'P5.14.1 DFT(k) of x(n)')set(gcf,'Color','white'); subplot(2,1,1); stem(k1, magXk_DFT); hold on; plot(N1*w/(2*pi), magXw_DTFT,'r--'); hold off;%axis([-N/2, N/2, -0.5, 50.5]);xlabel('k'); ylabel('magnitude(k)');title('DFT magnitude of x(n), N=12');  grid on;subplot(2,1,2); stem(k1, angXk_DFT);  hold on; plot(N1*w/(2*pi), angXw_DTFT,'r--'); hold off;%axis([-N/2, N/2, -0.5, 50.5]);xlabel('k'); ylabel('angle(k)');title('DFT angle of x(n), N=12');  grid on;
     

　　运行结果：

       原始序列长度为N=12，是4的倍数，这里v=3。

           从图看出，k为4的倍数时，X(k)为非零值。

           第2小题类似，这里直接上图：