Chapter 15: Multirate Digital Signal Processing

严肃小白兔

于 2025-04-01 20:00:00 发布

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分类专栏： DSP 文章标签： DSP Sampling

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Overview

This chapter introduces multirate DSP, which involves changing the sampling rate of digital signals. It is an essential technique used in many areas such as telecommunications, audio signal processing, and data compression. The chapter covers decimation, interpolation, sample rate conversion, and pitch shifting.

Decimation

Decimation is the process of reducing the sampling rate of a discrete-time signal by an integer factor M. This is done by retaining every M-th sample of the signal and discarding the rest.

Mathematical Expression

If the original signal is $x[n]$ , the decimated signal is:

$y[n]=x[nM]$

This operation alone is not safe if the signal contains frequency components above half of the new (lower) sampling rate. Those frequencies will alias (fold back into the band of interest), corrupting the signal.

Aliasing Problem

As stated in the document, “if the original signal contains frequency components above half the new sampling rate, these will alias into the baseband.”

Figure 1. Original Signal (top) and Downsampled Outcome (by a factor of 2) (bottom) [1].

As shown in Figure 1, since the Nyquist frequency is 5 kHz, the harmonic that was formerly at 6 kHz is reflected around this point, producing an aliased harmonic at 4 kHz [1].

Document Insight:
“It is therefore vital that decimation is always preceded by a low-pass filter that suppresses all frequencies beyond half the new sampling rate. This is called an anti-aliasing filter.”

Figure 2. Correctly Pre-filtered and Decimated [1].

the signal shown in Figure 2 shows the signal correctly decimated, since a new antialiasing filter was applied prior to downsampling.

Example (from the text)

The document presents an example where decimation is used in a digital audio system to reduce the sampling rate from 96 kHz to 48 kHz (M = 2). Before reducing the rate, a low-pass FIR filter is applied to eliminate frequencies above 24 kHz.

From the document:
“Decimation is not just a mathematical trick — it’s a necessity in real-time DSP when different parts of a system operate at different sample rates.”

Anti-Aliasing Filter

This is a low-pass filter used to remove unwanted high-frequency components before decimation.

Cut-off frequency: $f_c = \frac{f_s}{2M}$

Where $f_c$ is the cutoff frequency of the anti-aliasing filter. $f_s$ is the original sampling frequency (before decimation). $M$ is the decimation factor. It ensures that no signal content above the Nyquist frequency of the decimated signal gets through.