Case study of Interpolation and Decimation

THEORY

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

A variety of techniques have been developed to efficiently represent speech

signals in digital form for either transmission or storage. Since most of the speech

energy is contained in the lower frequencies, we would like to encode the lower frequency
band in more bits than the high-frequency band. Sub-band coding is a
method where the speech signal is subdivided into several frequency bands and
each band is digitally encoded separately.

 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 ”

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.

Basics
 

An example of a frequency subdivision is shown in the Figure 1. Let us
assume that the speech signal is sampled at a rate Fs samples per second. The first
frequency subdivision splits the signal spectrum into two equal width segments, a
low pass signal (0 < F < Fs/4) and a high pass signal (Fs/4 < F < Fs/2).
The second frequency subdivision splits the lowpass signal from the first
stage into two equal bands, a low pass signal (0 < F < Fs/8) and a high pass signal
(Fs/8 < F < Fs/4). Finally, the third frequency subdivision splits the low pass signal
from the second stage into two equal bandwidth signals. Thus, the signal is
subdivided into 4 frequency bands, covering 3 octaves, as shown in the figure.
Decimation by a factor of 2 is performed after frequency subdivision. By
allocating a different number bits per samples to the signals in the 4 sub-band, we
can achieve a reduction in the bitrate of the digitalized speech signal.

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Frequency Subdivision
Block Diagram: Sub band Speech Coder
Synthesis of Sub band Encoded Signal
Sub band coding of speech signal or application of Multirate processing in
DSP: simply we are decimating the signal i.e. decreasing the sampling rate at the
Transmitter, As all of us know Bandwidth Constraint is the main concern in Digital
Signal Processing. By decimating the signal at the Transmitter; it can be transmitted
with less number of Bits: ultimately results in saving the Bandwidth. 
While at the time of Reconstruction the signal is interpolated by the same factor, to achieve the original one.
The system at Transmitter and Receiver (Encoder and Decoder can be implemented using DSP Processor TMS 320 C 6713)

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Akash Bais is the Founder of EntcEngg and a passionate blogger. He loves new Technology, Tools, and Gadgets. He generally covers Technical, Industrial, and Job oriented aspects, etc in his posts.

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