Multi-rate Signal Processing - Interpolation and Interpolation Filters

Jul 14, 2025#数字通信7

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Interpolation, or upsampling, increases the sampling rate of a signal. Given an original sequence $ x(n) $ with sampling rate $ f_x $ and an interpolation factor $ L $, the new sequence $ y(m) $ is formed by inserting $ L-1 $ zeros between each pair of samples. The relationship between the sampling rates is $ f_y = L \times f_x $. In the frequency domain, the spectrum of the original sequence is periodically extended with period $ f_x $, while the new sequence's spectrum is extended with the new sampling rate $ f_y $. Although the frequency components remain unchanged, the spectrum at integer multiples of $ f_x $ creates mirror components, necessitating a low-pass filter to eliminate these artifacts. Since interpolation introduces zeros, it can alter the signal's amplitude, leading to potential amplitude loss. To maintain consistent amplitude before and after interpolation, a gain factor of $ 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 referred to as 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 becomes 0, it will alter the signal amplitude, thus causing a loss of signal amplitude. To ensure uniformity of signal amplitude before and after interpolation, a gain factor L can be set after the interpolation filter.