- DSP - Miscellaneous Signals
- DSP - Classification of DT Signals
- DSP - Classification of CT Signals
- DSP - Basic DT Signals
- DSP - Basic CT Signals
- DSP - Signals-Definition
- DSP - Home
Operations on Signals
- Operations Signals - Convolution
- Operations Signals - Integration
- Operations Signals - Differentiation
- Operations Signals - Reversal
- Operations Signals - Scaling
- Operations Signals - Shifting
Basic System Properties
- DSP - Solved Examples
- DSP - Unstable Systems
- DSP - Stable Systems
- DSP - Time-Variant Systems
- DSP - Time-Invariant Systems
- DSP - Non-Linear Systems
- DSP - Linear Systems
- DSP - Anti-Causal Systems
- DSP - Non-Causal Systems
- DSP - Causal Systems
- DSP - Dynamic Systems
- DSP - Static Systems
Z-Transform
- Z-Transform - Solved Examples
- Z-Transform - Inverse
- Z-Transform - Existence
- Z-Transform - Properties
- Z-Transform - Introduction
Discrete Fourier Transform
- DFT - Solved Examples
- DFT - Discrete Cosine Transform
- DFT - Sectional Convolution
- DFT - Linear Filtering
- DTF - Circular Convolution
- DFT - Time Frequency Transform
- DFT - Introduction
Fast Fourier Transform
Digital Signal Processing Resources
Selected Reading
- Who is Who
- Computer Glossary
- HR Interview Questions
- Effective Resume Writing
- Questions and Answers
- UPSC IAS Exams Notes
DSP - In-Place Computation
This efficient use of memory is important for designing fast hardware to calculate the FFT. The term in-place computation is used to describe this memory usage.
Decimation in Time Sequence
In this structure, we represent all the points in binary format i.e. in 0 and 1. Then, we reverse those structures. The sequence we get after that is known as bit reversal sequence. This is also known as decimation in time sequence. In-place computation of an eight-point DFT is shown in a tabular format as shown below −
POINTS | BINARY FORMAT | REVERSAL | EQUIVALENT POINTS |
---|---|---|---|
0 | 000 | 000 | 0 |
1 | 001 | 100 | 4 |
2 | 010 | 010 | 2 |
3 | 011 | 110 | 6 |
4 | 100 | 001 | 1 |
5 | 101 | 101 | 5 |
6 | 110 | 011 | 3 |
7 | 111 | 111 | 7 |
![Decimation in Time Sequence](/digital_signal_processing/images/decimation_in_time_sequence.jpg)
Decimation in Frequency Sequence
Apart from time sequence, an N-point sequence can also be represented in frequency. Let us take a four-point sequence to understand it better.
Let the sequence be $x[0], x[1], x[2], x[3], x[4], x[5], x[6], x[7]$. We will group two points into one group, initially. Mathematically, this sequence can be written as;
$$x[k] = sum_{n = 0}^{N-1}x[n]W_N^{n-k}$$Now let us make one group of sequence number 0 to 3 and another group of sequence 4 to 7. Now, mathematically this can be shown as;
$$displaystylesumpmits_{n = 0}^{frac{N}{2}-1}x[n]W_N^{nk}+displaystylesumpmits_{n = N/2}^{N-1}x[n]W_N^{nk}$$Let us replace n by r, where r = 0, 1 , 2….(N/2-1). Mathematically,
$$displaystylesumpmits_{n = 0}^{frac{N}{2}-1}x[r]W_{N/2}^{nr}$$We take the first four points (x[0], x[1], x[2], x[3]) initially, and try to represent them mathematically as follows −
$sum_{n = 0}^3x[n]W_8^{nk}+sum_{n = 0}^3x[n+4]W_8^{(n+4)k}$
$= lbrace sum_{n = 0}^3x[n]+sum_{n = 0}^3x[n+4]W_8^{(4)k} brace imes W_8^{nk}$
now $X[0] = sum_{n = 0}^3(X[n]+X[n+4])$
$X[1] = sum_{n = 0}^3(X[n]+X[n+4])W_8^{nk}$
$= [X[0]-X[4]+(X[1]-X[5])W_8^1+(X[2]-X[6])W_8^2+(X[3]-X[7])W_8^3$
We can further break it into two more parts, which means instead of breaking them as 4-point sequence, we can break them into 2-point sequence.
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