signal-processing - 如何将低通滤波器转换为带通滤波器

Question

我有一个由以下传递函数描述的低通滤波器：

h[n] = (w_c/Pi) * sinc( n * w_c / Pi )，其中 w_c 是截止频率

我必须将此低通滤波器转换为带通滤波器。

Answer 1

您h[n]转换到rect频域中。要使其带通，您需要将其中心频率移得更高。

要做到这一点，乘以h[n]，exp(j*w_offset*n)其中w_offset是移位量。如果w_offset是正数，那么你会转向更高的频率。

时域中的乘法是频域中的卷积。由于exp(j*w_offset*n)变成以 为中心的脉冲函数w_offset，因此乘法移动H(w)了w_offset。

有关详细信息，请参阅离散时间傅里叶变换。

注意：这样的过滤器不会关于 0 对称，这意味着它将具有复数值。为了使其对称，您需要添加h[n]乘以exp(-j*w_offset*n)：

h_bandpass[n] = h[n](exp(j*w_offset*n)+exp(-j*w_offset*n))

因为cos(w*n) = (exp(j*w*n)+exp(-j*w*n))/2我们得到：

h_bandpass[n] = h[n]cos(w_offset*n)

然后，该过滤器具有纯实值。

Answer 2

简短的回答是，您将乘以时域中的复指数。时域中的乘法将在频域中移动信号。

Matlab代码：

n_taps = 100;
n = 1:n_taps;
h = ( w_c / Pi ) * sinc( ( n - n_taps / 2) * w_c / Pi ) .* ...
    exp( i * w_offset * ( n - n_taps / 2) );

---

ps 我恰好在几周前刚刚为学校实现了这个确切的功能。

这是使用窗口方法创建自己的带通滤波器的代码：

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Function: Create bandpass filter using windowing method
% Purpose:  Simple method for creating filter taps ( useful when more elaborate
%           filter design libraries are not available )
%
% @author   Trevor B. Smith, 24MAR2009
%
% @param    n_taps    How many taps are in your output filter
% @param    omega_p1  The lower cutoff frequency for your passband filter
% @param    omega_p2  The upper cutoff frequency for your passband filter
% @return   h_bpf_hammingWindow     The filter coefficients for your passband filter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function h_bpf_hammingWindow = BPF_hammingWindow(n_taps,omega_p1,omega_p2)
    % Error checking
    if( ( omega_p2 == omega_p1 ) || ( omega_p2 < omega_p1 ) || ( n_taps < 10 ) )
        str = 'ERROR - h_bpf_hammingWindow():   Incorrect input parameters'
        h_bpf_hammingWindow = -1;
        return;
    end

    % Compute constants from function parameters
    length = n_taps - 1; % How many units of T ( i.e. how many units of T, sampling period, in the continuous time. )
    passbandLength = omega_p2 - omega_p1;
    passbandCenter = ( omega_p2 + omega_p1 ) / 2;
    omega_c = passbandLength / 2; % LPF omega_c is half the size of the BPF passband
    isHalfSample = 0;
    if( mod(length,2) == 1 )
        isHalfSample = 1/2;
    end

    % Compute hamming window
    window_hamming = hamming(n_taps);

    % Compute time domain samples
    n = transpose(-ceil(length/2):floor(length/2));
    h1 = sinc( (1/pi) * omega_c * ( n + isHalfSample ) ) * pi .* exp( i * passbandCenter * ( n + isHalfSample ) );

    % Window the time domain samples
    h2 = h1 .* window_hamming;
    if 1
        figure; stem(h2); figure; freqz(h2);
    end

    % Return filter coefficients
    h_bpf_hammingWindow = h2;
end % function BPF_hammingWindow()

如何使用此功能的示例：

h_bpf_hammingWindow = BPF_hammingWindow( 36, pi/4, 3*pi/4 );
freqz(h_bpf_hammingWindow); % View the frequency domain

Answer 3

让f[n]您从低通滤波器获得的信号w_c位于所需频带的下限。f[n]您可以通过从原始信号中减去来获得高于此下限的频率。这是您想要用于第二个低通滤波器的输入。