Downsampling and Upsampling: Time and Frequency Domain Analysis

1. Decimation: Reducing the Sampling Rate

Decimation is the process of decreasing the sampling rate of a discrete-time signal. In the previous lesson, we saw this was crucial for converting oversampled signals back to a standard rate. The process consists of two key steps, as shown in the block diagram below.

Figure 1: Block diagram of decimation by a factor of M. Adapted from "Chapter 9 – Multirate Digital Signal Processing".

1. Anti-Aliasing Filtering: The input signal $x[n]$ is first passed through a digital low-pass filter.
2. Compression: The filtered signal, $v[n]$, is then downsampled by keeping every $M$-th sample and discarding the $M-1$ samples in between. This operation is defined as:
   $y[n]=v[nM]$

Time-Domain Perspective

In the time domain, the compression step is straightforward. For a decimation factor of $M=3$, we keep samples at indices 0, 3, 6, ... and discard samples at 1, 2, 4, 5, 7, 8, ... The new sequence $y[n]$ is formed by re-indexing the kept samples: $y[0]=v[0]$, $y[1]=v[3]$, $y[2]=v[6]$, and so on.

The critical question is: why is the filter necessary? To answer that, we must turn to the frequency domain.

Frequency-Domain Analysis

Changing the sampling rate of a signal has a profound effect on its spectrum. The following video provides a clear and detailed mathematical derivation of what happens when a signal is downsampled.

As the video explains, if a signal $y[n]$ is created by downsampling a signal $v[n]$ by a factor of $M$, i.e., $y[n]=v[nM]$, their Discrete-Time Fourier Transforms (DTFTs) are related by:

Let's break down this important formula:

1. Frequency Scaling (Stretching): The term $V\left(e^{j\omega /M}\right)$ (when $k=0$) shows that the original spectrum is stretched along the frequency axis by a factor of $M$. A frequency ${\omega }_0$ in $V(e^{j\omega })$ moves to $M{\omega }_0$ in $Y(e^{j\omega })$.
2. Spectral Copies (Aliasing): The summation from $k=0$ to $M-1$ creates $M-1$ additional copies of the stretched spectrum, shifted by multiples of $2\pi$, which are then summed together.

This is where the anti-aliasing filter becomes essential. If the original signal $x[n]$ has frequency components above $\pi /M$, the "stretching" effect will cause these high frequencies to fold back into the $[-\pi ,\pi ]$ interval of the new spectrum, overlapping with the desired baseband spectrum. This is aliasing.

The job of the digital anti-aliasing filter $h[k]$ is to ensure that the signal $v[n]$ entering the downsampler has no energy above the cutoff frequency ${\omega }_c=\pi /M$. By removing these frequencies beforehand, we guarantee that when the spectrum is stretched, no aliasing occurs.

Figure 2: The effect of decimation on the frequency spectrum. The anti-aliasing filter (b) is crucial for preventing the spectral overlap (aliasing) seen in (c). Adapted from "Chapter 9 – Multirate Digital Signal Processing".

2. Interpolation: Increasing the Sampling Rate

Interpolation is the process of increasing the sampling rate. It's essentially the inverse of decimation and is used, for example, in oversampling digital-to-analog converters to simplify the analog reconstruction filter.

The process involves two steps:

1. Expansion (Upsampling): First, we create a new signal $w[n]$ by inserting $L-1$ zeros between every sample of the original signal $x[n]$. This operation is sometimes called a "stretcher" or "expander."
   $w[n]=\{\begin{array}{ll}x[n/L],& \text{if }n\text{ is a multiple of }L\\ 0,& \text{otherwise}\end{array}$
2. Interpolation Filtering: The zero-stuffed signal $w[n]$ is then passed through a digital low-pass filter. This filter "fills in" the zero values with properly interpolated samples and is often called an anti-imaging filter.

Figure 3: Block diagram of interpolation by a factor of L. Adapted from "Chapter 9 – Multirate Digital Signal Processing".

Frequency-Domain Analysis

Let's see how these two steps affect the signal's spectrum. The following video derives the frequency-domain relationships for the interpolation process.

As derived in the video, the two steps have distinct effects on the spectrum:

1. Expansion (Zero-Insertion): The DTFT of the zero-stuffed signal, $W(e^{j\omega })$, is related to the original signal's DTFT, $X(e^{j\omega })$, by:
   $W(e^{j\omega })=X(e^{j\omega L})$
   This means the frequency axis is compressed by a factor of $L$. The original spectrum from $[-\pi ,\pi ]$ is squeezed into the range $[-\pi /L,\pi /L]$. Because the DTFT is always $2\pi$-periodic, this compression results in $L-1$ additional copies of the spectrum appearing within the $[-\pi ,\pi ]$ interval. These unwanted copies are called spectral images.

2. Anti-Imaging Filtering: The purpose of the low-pass filter is to remove these unwanted images. The filter needs to have a cutoff frequency of ${\omega }_c=\pi /L$, preserving the original baseband spectrum and eliminating everything else. Furthermore, the act of inserting zeros reduces the total energy of the signal. To compensate for this, the filter must have a gain of $L$ in its passband to restore the signal's original amplitude.

The combined result is a signal $y[n]$ with a higher sampling rate, whose spectrum is a correctly scaled version of the original spectrum, but now occupying a smaller fraction of the available $[-\pi ,\pi ]$ bandwidth.

Figure 4: The effect of interpolation on the frequency spectrum. The anti-imaging filter removes the spectral images created by the expansion process. Adapted from "Chapter 9 – Multirate Digital Signal Processing".

For a more condensed, mathematically rigorous summary of both decimation and interpolation using z-transform notation, the "Upsampling and Downsampling" lecture notes from Stanford are an excellent resource.

Conclusion

Today, we have established the fundamental mathematical principles of decimation and interpolation. By analyzing these processes in both the time and frequency domains, we've uncovered the precise mechanisms of spectral stretching, aliasing, compression, and imaging.

Key Takeaways:

• Decimation by a factor $M$ involves low-pass filtering with a cutoff of $\pi /M$ and then discarding $M-1$ out of every $M$ samples. In the frequency domain, this corresponds to stretching the spectrum by $M$. The filter is essential to prevent aliasing.
• Interpolation by a factor $L$ involves inserting $L-1$ zeros between samples and then low-pass filtering. In the frequency domain, zero-insertion compresses the spectrum by $L$ and creates $L-1$ spectral images.
• The anti-imaging filter in an interpolator has two jobs: remove the unwanted spectral images (with a cutoff of $\pi /L$) and restore the signal's amplitude (with a passband gain of $L$).

Preview of the Next Lesson:

We now understand what decimation and interpolation do to a signal. However, we also know from our first lesson that performing these operations naively (e.g., filtering at the high rate before decimating) is computationally wasteful. In our next lesson, we will finally bridge this gap by deriving the polyphase representation of FIR filters. This elegant mathematical structure allows us to rearrange the order of operations, leading to massive gains in computational efficiency and revealing the true power of multirate signal processing.