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] 转换为频域中的 rect 。为了使它成为带通，你需要将其中心频率移动得更高。

为此，将 h [n] 乘以 exp（j * w_offset * n），其中 w_offset 是要转移的金额。如果 w_offset 为正，则转向更高的频率。

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

有关详细信息，请参阅离散时间傅里叶变换。

注意：这样的过滤器不会对称于0，这意味着它将具有复杂的值。要使其对称，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））

由于 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）

此过滤器具有纯粹的实际值。

Answer 2

简短的回答是，您将在时域中乘以复指数。时域中的乘法将在频域中移位信号。

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) );

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P.S。几周前，我碰巧刚为学校实施了这个功能。

以下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] 成为低通滤波器的信号，其中 w_c 位于所需频段的下限。您可以通过从原始信号中减去 f [n] 来获得高于此下限的频率。这是第二个低通滤波器所需的输入。