Sampling:Decimation and Interpolation

Sampling:

 

Sampling is the process of representing a continuous signal with a sequence of discrete data values. In practice, sampling is performed by applying a continuous signal to an analog-to-digital (A/D) converter whose output is a series of digital values

image

 

Sampling can be viewed theoretically as multiplying the original signal with a train of pulses with unit amplitude. This leaves the original signal with information at discrete points with magnitude of signal present in the original signal at that position. This is illustrated in the figure below.

 

 

image

 Downsampling (Decimation):

 

Decimation is a technique for reducing the number of samples in a discrete-time signal.

The operation of downsampling by factor M describes the process of keeping every Mth sample and discarding the rest. This is denoted by ” ↓M ↓ M ” in block diagrams

 

image

 

The decimation process filters the input data with a low pass filter and then resamples the resulting smoothed signal at a lower rate. image

 

Decimation is the process of filtering and downsampling a signal to decrease its effective sampling rate, as illustrated in figure. The filtering is employed to prevent aliasing that might otherwise result from downsampling.

image
Why Decimate?

 The most immediate reason to decimate is simply to reduce the sampling rate at the output of one system so a system operating at a lower sampling rate can input the signal. But a much more common motivation for decimation is to reduce the cost of processing: the calculation and/or memory required to implement a DSP system generally is proportional to the sampling rate, so the use of a lower sampling rate usually results in a cheaper implementation.

 

Upsampling (Interpolation):

Interpolation is a technique for increasing the number of samples in a discrete-time signal. The operation of upsampling by factor L describes the insertion of L-1 L 1 zeros between every sample of the input signal. This is denoted by “↑L ↑ L “in block diagrams, as in figure.

 

image

The process of increasing the sampling rate is called interpolation. Interpolation is upsampling followed by appropriate filtering.  obtained by interpolating  is generally represented as:

image

The simplest method to interpolate by a factor of L is to add L-1 zeros in between the samples multiply the amplitude by L and filter the generated signal, with a so-called anti-imaging low pass filter at the high sampling frequency.

 

image

 Different Methods of Interpolation:

1) Nearest neighbor interpolation:

This method sets the value of an interpolated point to the value of the nearest existing data point.

2) Linear interpolation:

This method fits a different linear function between each pair of existing data points, and returns the value of the relevant function at the points specified.

3) Cubic interpolation:

This method fits a different cubic function between each pair of existing data points

Purpose of Interpolation:

These processes are required to make the sampling rate of the signal compatible with the bandwidth of the signal processing system. Integer factor interpolation and decimation algorithms may be implemented using efficient FIR filters.

Let's Discuss