Subband Coding, Filter Banks, and Narrowband Design

Applications of Multirate DSP (Proakis 11.9)

1. Subband Coding (Speech/Audio Compression)

Input ──► Analysis Filter Bank ──► Quantize/Encode ──► Bitstream
         (Split into subbands)    (Allocate bits per subband)

• Used in: MP3, AAC, Dolby AC-3
• Perceptual coding: More bits to important subbands

2. Quadrature Mirror Filters (QMF) / Perfect Reconstruction

Two-channel filter bank:

         H_0(z) (LP)          G_0(z)
x[n] ──►───────►───►[↓2]──►───►───────►
         H_1(z) (HP)          G_1(z)
          └───────►───►[↓2]──►───────►
                └────────► Sum ◄─────┘

Perfect reconstruction if $G_0(z)H_0(z) + G_1(z)H_1(z) = 2z^{-k}$

3. Digital Filter Banks (Channelization)

• Split wideband signal into narrow subbands
• Process each independently (beamforming, spectrum monitoring)
• Efficient via polyphase FFT filter banks

4. Interfacing Systems with Different Rates

• CD (44.1k) → DAT (48k)
• ADC at high rate → process at low rate → DAC at high rate
• Oversampling ADCs/DACs

5. Narrowband Lowpass Filter Implementation

Very narrow transition width → high-order filter Multirate approach:

1. Decimate by $M$ (reduce rate)
2. Design wider filter at lower rate
3. Interpolate back

$M$-fold reduction in filter order!

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Design Example: Narrowband Lowpass via Multirate

Specs: $F_s = 48$ kHz, passband 0–100 Hz, stopband > 200 Hz, $A_s = 80$ dB Direct FIR: $\Delta f = 100$ Hz → $N \approx 4 \cdot 48000/100 = 1920$ taps!

Multirate ($M=100$):

1. Decimate by 100 → $F_s’ = 480$ Hz
2. New specs: passband 0–100 Hz, stopband > 200 Hz at $F_s’ = 480$ Hz
   • $\omega_p = 100/240 \pi = 0.417\pi$
   • $\omega_s = 200/240 \pi = 0.833\pi$
3. Design filter at low rate → $N’ \approx 20$ taps
4. Interpolate by 100 Total work: ~100× fewer multiplies!

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MATLAB Implementation

% Decimation
M = 4;
x = randn(1, 1000);
h = fir1(50, 1/M);        % Anti-aliasing filter
y = decimate(x, M);        % Built-in (filters + downsamples)

% Manual polyphase decimation
h_poly = reshape(h, M, []).';  % M branches
y_poly = zeros(1, length(x)/M);
for k = 1:M
    y_poly = y_poly + filter(h_poly(k,:), 1, x(k:M:end));
end

% Interpolation
L = 3;
y_interp = interp(x, L);    % Built-in (upsamples + filters)

% Rational rate conversion
L = 160; M = 147;
y_resamp = resample(x, L, M);  % Single function!

% Polyphase filter bank (analysis)
M = 32;  % 32 channels
h_proto = fir1(256, 1/M);       % Prototype LPF
% Modulate to create filter bank
for k = 0:M-1
    H_bank(:,k+1) = h_proto .* exp(1j*2*pi*k*(0:length(h_proto)-1)/M);
end

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