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Linear and cyclic convolution

Lecture



Content
Introduction
Linear convolution
Cyclic convolution
Algorithm for fast calculation of convolution based on FFT
findings
Introduction
One of the radio engineering whales is undoubtedly the convolution operation:
Linear and cyclic convolution (one)
Convolution allows you to calculate the signal Linear and cyclic convolution at the output of the linear filter with impulse response Linear and cyclic convolution , at the input signal Linear and cyclic convolution .
In the discrete case, there are two types of convolutions: linear (or aperiodic) and cyclic. Cyclic convolution is often referred to as circular or periodic.
Linear convolution
Consider a linear convolution. Let there be two discrete signals. Linear and cyclic convolution , Linear and cyclic convolution and Linear and cyclic convolution , Linear and cyclic convolution . In general, the length of these signals Linear and cyclic convolution and Linear and cyclic convolution may differ. Linear convolution of signals Linear and cyclic convolution and Linear and cyclic convolution called a discrete signal of the form:
Linear and cyclic convolution (2)
To calculate linear convolution signals Linear and cyclic convolution and Linear and cyclic convolution shift relative to each other termwise multiply and fold. It is assumed that Linear and cyclic convolution at Linear and cyclic convolution and Linear and cyclic convolution , and Linear and cyclic convolution at Linear and cyclic convolution and Linear and cyclic convolution
A graphical representation of linear convolution is presented in Figure 1.
Linear and cyclic convolution
Figure 1: Graphical representation of linear convolution
Signal counts Linear and cyclic convolution shift relative to the sequence counts Linear and cyclic convolution all possible overlapping counts multiply together and add up.
Figure 2 shows an example of calculating the linear convolution of two signals. Linear and cyclic convolution 4 counts long Linear and cyclic convolution 3 counts long.
Linear and cyclic convolution
Figure 2: Example of a linear convolution calculation.
It should be noted that the signal Linear and cyclic convolution when calculating the convolution is reflected from left to right, because Linear and cyclic convolution the very first count (the earliest in time) and it should also be processed first.
Cyclic convolution
We now consider cyclic convolution. In the case of cyclic convolution, it is assumed that discrete signals Linear and cyclic convolution and Linear and cyclic convolution - periodic with the same period Linear and cyclic convolution counts. Then circular convolution of the signals Linear and cyclic convolution and Linear and cyclic convolution called the signal type:
Linear and cyclic convolution (3)
The result of cyclic convolution also has a length Linear and cyclic convolution counts.
Consider a cyclic convolution using the example of two signals. Linear and cyclic convolution and Linear and cyclic convolution . Graphically, the calculation of cyclic convolution is presented in Figure 3.
Linear and cyclic convolution
Figure 3: Calculating the circular convolution
The red line marks the boundaries of the repetition periods. Linear and cyclic convolution . Note that due to the periodicity of the signals Linear and cyclic convolution .
Calculate the convolution step by step:
Linear and cyclic convolution (four)
Now let's calculate Linear and cyclic convolution :
Linear and cyclic convolution (five)
Similarly, you can calculate Linear and cyclic convolution and Linear and cyclic convolution .
Using cyclic convolution, one can calculate the linear convolution of two signals. This requires each of the signals Linear and cyclic convolution and Linear and cyclic convolution duration Linear and cyclic convolution and Linear and cyclic convolution counts respectively add zeros to length Linear and cyclic convolution .
We give an example of calculating a linear convolution through a cyclic for Linear and cyclic convolution 4 counts long Linear and cyclic convolution 3 references long (this example was considered above).
We add zeros Linear and cyclic convolution and Linear and cyclic convolution , so that in each sequence was 6 samples.
Calculate the cyclic convolution as shown in Figure 4.
Linear and cyclic convolution
Figure 4: Calculating linear convolution through cyclic <
You can compare with the result of the very first example for linear convolution and make sure that the values ​​match.
Algorithm for fast calculation of convolution based on FFT
At first glance it may seem that the calculation of linear convolution through cyclic does not make sense, since it does not reduce the calculation. Valid to calculate linear convolution required Linear and cyclic convolution multiplications Linear and cyclic convolution and Linear and cyclic convolution - the length of the convolved signals, and when calculating a linear convolution through a cyclic Linear and cyclic convolution multiplications and additions. However, consider the discrete Fourier transform of cyclic convolution:
Linear and cyclic convolution (6)
Substituting the expression for cyclic convolution we get:
Linear and cyclic convolution (7)
Swap the summation operations:
Linear and cyclic convolution (eight)
Imagine a multiplier Linear and cyclic convolution as:
Linear and cyclic convolution (9)
Substituting (9) into (8) we get:
Linear and cyclic convolution (ten)
Thus, the cyclic convolution spectrum is the product of the spectra of the convolved signals, and the Fast Fourier Transform (FFT) algorithm can be used to calculate it. Thus, the circuit for calculating cyclic convolution can be represented in Figure 5:
Linear and cyclic convolution
Figure 5: Calculating cyclic convolution using FFT
Since effective FFT algorithms do not exist for all lengths Linear and cyclic convolution then we can propose the following method for calculating cyclic convolution. Source sequences Linear and cyclic convolution and Linear and cyclic convolution , Linear and cyclic convolution can be filled with zeros to length Linear and cyclic convolution Since for effective lengths of an integer power of two, efficient FFT algorithms have been developed, then we apply the convolution calculation scheme presented in Figure 5. At the output, we obtain Linear and cyclic convolution , Linear and cyclic convolution from which you need to take only the first Linear and cyclic convolution counts.
Similarly, you can do when calculating linear convolution through cyclic.
Consider an example. Let be Linear and cyclic convolution , but Linear and cyclic convolution . Direct linear convolution calculation will require Linear and cyclic convolution (12 million) multiplication and addition operations.
Fill in each of the sequences up to 8192 samples with zeros and apply the FFT algorithm with decimation by time, then the calculation of one FFT will be required Linear and cyclic convolution complex multiplication operations or 428000 real multiplication operations. There will be only 3 such blocks of FFT, plus you need to take into account 8192 complex multiplications of the spectra, total Linear and cyclic convolution , which is almost 7.5 times lower than if we considered linear convolution in the forehead.
findings
Thus, we considered two types of discrete convolutions: linear and cyclic; we established a connection between them. It has been shown that the use of FFT provides a significant reduction in computational operations when calculating both cyclic and linear convolutions.
created: 2015-05-23
updated: 2022-01-09
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Digital signal processing

Terms: Digital signal processing