Decimator and Interpolator Relationship in Signal Processing

What is the relationship between a decimator and an interpolator in signal processing?

In signal processing, how does the transpose of a factor-of-M decimator relate to a factor-of-M interpolator?

Answer:

The relationship between a decimator and an interpolator in signal processing is crucial for understanding how different operations affect the sample rate of a signal. When we examine the transpose of a factor-of-M decimator and its connection to a factor-of-M interpolator, we gain insights into the inverse operations of decimation and interpolation.

In signal processing, a decimator and an interpolator are essential tools used to modify the sample rate of a signal. Decimation is the process of reducing the sample rate, typically by discarding samples, while interpolation is the process of increasing the sample rate, usually by inserting new samples.

A factor-of-M decimator reduces the sample rate by a factor of M by discarding M-1 out of every M samples. This operation is denoted as D(M). On the other hand, a factor-of-M interpolator increases the sample rate by a factor of M by adding interpolated samples between original samples.

When we consider the transpose of a factor-of-M decimator, denoted as D(M)ᵀ, it performs the opposite operation of the decimator. Instead of reducing the sample rate, the transpose of the decimator increases the sample rate by a factor of M. It achieves this by adding M-1 zero-valued samples between each original sample, effectively interpolating the signal.

Therefore, the relationship between a factor-of-M decimator and a factor-of-M interpolator is that the transpose of a factor-of-M decimator acts as a factor-of-M interpolator. This relationship highlights how the operations of decimation and interpolation are interconnected and can be reversed using the transpose operation.