Multi-rate Signal Processing - Interpolation and Interpolation Filters

Sep 23, 2024#数字通信73

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Interpolation, also known as upsampling, involves increasing the sampling rate of a signal. Given an original sequence $ x(n) $ with a sampling rate $ f_x $ and an interpolation factor $ L $, the process inserts $ L-1 $ zeros between each pair of adjacent samples to create a new sequence $ y(m) $. The relationship between the sampling rates is given by $ f_y = L \times f_x $. From a frequency domain perspective, the frequency spectrum of the original sequence is periodically extended with period $ f_x $. After interpolation, the new sequence's spectrum is extended with the new sampling rate $ f_y $. While the spectral components remain unchanged, the frequency points at integer multiples of $ f_x $ introduce mirror components. To eliminate these, a low-pass filter is necessary, typically implemented with an upsampler and an anti-aliasing filter. Since interpolation involves inserting zeros, it can alter the signal's amplitude, leading to amplitude loss. To maintain consistent amplitude before and after interpolation, a gain factor $ L $ can be applied after the interpolation filter.

Interpolation means increasing the sampling rate, hence it is called upsampling. Let the original sequence be $x(n)$ , the sampling rate be $f_x$ , and the interpolation factor be $L$ . The interpolation process involves inserting $L-1$ zeros between every two adjacent samples of the original sequence to form a new sequence, mathematically expressed as

y(m)=\begin{cases}x(m/L)&m=0,\pm L,\pm2L,\cdots\\0&\text{otherwise}\end{cases}

Let $f_y$ represent the sampling rate of $y(m)$ , then the relationship between the sampling rates is

f_y=L\times f_x

The illustration is as follows:

From the frequency domain perspective, the frequency spectrum of the original sequence is periodically extended with a period of $f_x$ .

The new sequence after interpolation is periodically extended with the new sampling rate $f_y$ .

It can be seen that the spectral components before and after interpolation remain unchanged, but the spectral components at integer multiples of $f_x$ are called mirror components. Therefore, a low-pass filter must be added after interpolation to eliminate the mirror frequency. A typical interpolator is completed by a combination of an upsampler and an anti-mirror filter.

Since interpolation involves inserting zero values into the original sequence, meaning that the signal amplitude at certain sampling points is $0$ , it will alter the signal amplitude, thus causing a loss of signal amplitude. To ensure the uniformity of signal amplitude before and after interpolation, a gain factor of $L$ can be set after the interpolation filter.