Multirate DSP: Decimation and Interpolation

Why Multirate?

Different subsystems need different sampling rates:

• Audio CD: 44.1 kHz
• Telephony: 8 kHz
• Professional audio: 48/96/192 kHz
• Video: MHz–GHz range

Multirate DSP changes sampling rate entirely in digital domain—no D/A → A/D conversion needed.

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Decimation (Down-Sampling) by Factor $M$

Process:

x[n] ──► [Lowpass Filter H(z)] ──► [↓M] ──► y[n]
           (Anti-aliasing)           (Keep every M-th)

Steps:

1. Filter: Lowpass with cutoff $\omega_c = \pi/M$ to prevent aliasing
2. Down-sample: $y[n] = x[Mn]$

Frequency Domain:

\(Y(e^{j\omega}) = \frac{1}{M} \sum_{k=0}^{M-1} X(e^{j(\omega-2\pi k)/M})\)

Spectra stretch by $M$ and replicate. Filter removes replicas that would alias.

Example: Decimate by 2

• Input: $x[n]$ at $F_s$
• Filter: LPF cutoff $\pi/2$ (Nyquist of new rate)
• Output: $y[n] = x[2n]$ at $F_s/2$

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Interpolation (Up-Sampling) by Factor $L$

Process:

x[n] ──► [↑L] ──► [Lowpass Filter H(z)] ──► y[n]
        (Insert L-1 zeros)        (Anti-imaging)

Steps:

1. Up-sample: Insert $L-1$ zeros between samples \(v[n] = \begin{cases} x[n/L] & n \text{ multiple of } L \\ 0 & \text{otherwise} \end{cases}\)
2. Filter: Lowpass with cutoff $\pi/L$ to interpolate zeros → smooth signal
3. Gain: Multiply by $L$ to compensate for zero-insertion energy loss

Frequency Domain:

\(V(e^{j\omega}) = X(e^{j\omega L})\) Images appear at multiples of $2\pi/L$. Filter removes images.

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