Matlab中滤波器的逆FT

Question

我试图在 Matlab 中找到一个简单滤波器的傅里叶逆变换。在第一种情况下（sinc 滤波器/“砖墙”），我使用ifft 函数找到时域函数，它是一个 sinc，以 t = 0 为中心。

我现在想找到一个简单 Chebyshev 滤波器的时域函数。但是，由于某种原因，下面的代码似乎给了我脉冲响应，其中时间轴不正确。我不应该期待以 t = 0 为中心的类似外观的 sinc 函数吗？

fo = 10; %frequency of the sine wave
Fs = 100; %sampling rate
Ts = 1/Fs; %sampling time interval
t = -1+Ts:Ts:1-Ts; %sampling period
freq = -Fs/2:(Fs/length(t)):Fs/2;

%% Sinc with bandwidth = fo. This works!
y = 0.5*sinc(2*fo*t);
YfreqDomain1 = fft(y);
figure('Name','Brick wall sinc filter (freq)');
plot(freq(1:length(y)),2/length(y)*fftshift(abs(YfreqDomain1)))
y_ret1=ifft(YfreqDomain1,'nonsymmetric');
figure('Name','Brick wall sinc filter (time)');
plot(t,y_ret1);

%% Chebyshev with bandwidth fo. This gives me a strange result. 
[b,a] = cheby1(6,0.1,2*fo/Fs);  % 6th order, 0.1dB ripple
[YfreqDomain2 w] = freqz(b,a,length(t),'whole');
figure('Name','Chebyshev Filter (freq)');
plot(freq(1:length(YfreqDomain2)), 2/length(y)*fftshift(abs(YfreqDomain2)));
figure('Name','Chebyshev Filter (time)');
y_ret2=ifft(YfreqDomain2,'nonsymmetric');
plot(t,y_ret2);

Answer 1

您的计算存在两个问题：

首先，您在一个非常窄的时间窗口上评估时域滤波器系数，这会截断滤波器并改变其特性。

其次，您没有正确跟踪时域和频域向量中的哪些索引分别对应于时间点 0 和频率 0。这可以通过不同的方式完成，我在这里选择始终使用t(1) = 0 和f(1) = 0，并将fftshift 仅用于绘图。

这是您的代码的更正版本：

fo = 10;        % frequency of the sine wave
Fs = 100;       % sampling rate
Ts = 1 / Fs;    % sampling time interval
n = 1000;       % number of samples

% prepare sampling time vector such that t(1) = 0
t = (0 : n - 1)';
t = t - n * (t >= 0.5 * n);
t = t / Fs;

% prepare frequency vector such that f(1) = 0;
f = (0 : n - 1)' / n;
f = f - (f >= 0.5);
f = f * Fs;


%% sinc filter with bandwidth fo

% define filter in time domain
s = 2*fo/Fs * sinc(2*fo*t);

% transform into frequency domain
Hs = fft(s);
% since the filter is symmetric in time, it is purely real in the frequency
% domain. remove numeric deviations from that:
Hs = real(Hs);

subplot(2, 4, 1)
plot(fftshift(t), fftshift(s))
ylim([-0.1 0.25])
title('sinc, time domain')

subplot(2, 4, 2)
plot(fftshift(f), real(fftshift(Hs)), ...
    fftshift(f), imag(fftshift(Hs)))
ylim([-1.1 1.1])
title({'sinc, frequency domain', 'real / imaginary'})

subplot(2, 4, 3)
plot(fftshift(f), abs(fftshift(Hs)))
ylim([-0.1 1.1])
title({'sinc, frequency domain', 'modulus'})


%% Chebyshev filter with bandwidth fo

% define filter in frequency domain
[b,a] = cheby1(6, 0.1, 2*fo/Fs);  % 6th order, 0.1 dB ripple
Hc = freqz(b, a, n, 'whole', Fs);

% transform into time domain
c = ifft(Hc);

% determine phase such that phase(1) is as close to 0 as possible
phase = fftshift(unwrap(angle(fftshift(Hc))));
phase = phase - 2*pi * round(phase(1) /2/pi);

subplot(2, 4, 5)
plot(fftshift(t), fftshift(c))
title('Chebyshev, time domain')
ylim([-0.1 0.25])

subplot(2, 4, 6)
plot(fftshift(f), real(fftshift(Hc)), ...
    fftshift(f), imag(fftshift(Hc)))
ylim([-1.1 1.1])
title({'Chebyshev, frequency domain', 'real / imaginary'})

subplot(2, 4, 7)
plot(fftshift(f), abs(fftshift(Hc)))
ylim([-0.1 1.1])
title({'Chebyshev, frequency domain', 'modulus'})

subplot(2, 4, 8)
plot(fftshift(f), fftshift(phase))
title({'Chebyshev, frequency domain', 'phase'})

结果如下：

如您所见，sinc 和 Chebyshev 滤波器在频率响应的模数方面相似，但在相位方面却大不相同。原因是切比雪夫滤波器是causal filter，这意味着时域中的系数在 t