《DSP using MATLAB》Problem 2.20

代码：

%% ------------------------------------------------------------------------
%%            Output Info about this m-file
fprintf(‘\n***********************************************************\n‘);
fprintf(‘        <DSP using MATLAB> Problem 2.20 \n\n‘);

banner();
%% ------------------------------------------------------------------------

b = [1, -1]; a = [1];

%% -------------------------------------------------
%%           3 input sequence
%% -------------------------------------------------
n1 = [0:30];
x1 = 5 * (stepseq(0, 0, 30) - stepseq(20, 0, 30));

n2 = [-1:30];
x2 = n2 .* (stepseq(0, -1, 30)-stepseq(10, -1, 30)) + (20-n2) .* (stepseq(10, -1, 30)-stepseq(20, -1, 30));

n3= [-1:110];
x3 = sin(pi*n3/25) .* (stepseq(0, -1, 110)-stepseq(100, -1, 110));

%% ------------------------------------------------
%%            3 output sequence
%% ------------------------------------------------
y1 = filter(b, a, x1);
y2 = filter(b, a, x2);
y3 = filter(b, a, x3);

figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Problem 2.20‘)
set(gcf,‘Color‘,[1,1,1])                  % 改变坐标外围背景颜色

subplot(2,1,1); stem(n1, x1); title(‘x1‘);
xlabel(‘n‘); ylabel(‘x1(n)‘) ;
grid on
subplot(2,1,2); stem(n1, y1); title(‘y1‘);
xlabel(‘n‘); ylabel(‘y1(n)‘);
grid on;

figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Problem 2.20‘)
set(gcf,‘Color‘,[1,1,1])                  % 改变坐标外围背景颜色

subplot(2,1,1); stem(n2, x2); title(‘x2‘);
xlabel(‘n‘); ylabel(‘x2(n)‘) ;
grid on
subplot(2,1,2); stem(n2, y2); title(‘y2‘);
xlabel(‘n‘); ylabel(‘y2(n)‘);
grid on;

figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Problem 2.20‘)
set(gcf,‘Color‘,[1,1,1])                  % 改变坐标外围背景颜色

subplot(2,1,1); stem(n3, x3); title(‘x3‘);
xlabel(‘n‘); ylabel(‘x3(n)‘) ;
grid on
subplot(2,1,2); stem(n3, y3); title(‘y3‘);
xlabel(‘n‘); ylabel(‘y3(n)‘);
grid on;

%% ---------------------------------------------
%%       stability
%% ---------------------------------------------
x4 = impseq(0, 0, 10);
y4 = filter(b, a, x4);

fprintf(‘\nsum(abs(h)) = %f \n‘, sum(abs(y4)));

figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Problem 2.20 Zero-Pole‘)
set(gcf,‘Color‘,[1,1,1])                  % 改变坐标外围背景颜色
pzmap(b, a);

z = roots(a); magz = abs(z)

　　运行结果：

上图看出，三角波信号的微分结果是方波。正弦信号的微分当然就是余弦信号了，见下图：

微分器的脉冲响应序列是绝对可和的，所以系统是稳定的。