我正在尝试使用 Matlab 的filter函数求解具有初始条件的差分方程：

$$y[n]=\frac12y[n-1]+x[n],\qquad y[-1]=1\tag{1}$$

为了$x[n]=6u[n]$, 计算$y[n]$用手给

$$y[n]=\left[12-(6-y[-1]/2)\left(\frac12\right)^n\right]u[n]\tag{2}$$

但是，产生的解决方案filter是不同的。有人能指出我的错误吗？

代码如下所示：

disp('Solution of a difference equation:')
% The example difference equation considered is
% y(n) = 0.5y(n-1) + x(n) 
num_coef = [ 1 ]; % coefficient of x
den_coef = [1 -0.5]; % coefficient of y
n = 0:4; % Considering five samples 
x = 6 * ones(1,5) % x(n) = 6u(n)
init_cond = [1];  % y(-1) = 1
y = filter(num_coef, den_coef, x, init_cond) % Solution returned by 'filter'
yM = 12 - (11/2)*(1/2).^n % Solution obtained manually
plot(y);hold on; % PLots for comparison
plot(yM,'r')