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65ANALYSIS OF DISCRETE CONVOLUTION IN THE MATLAB PROGRAM
Mokhirakhon Kurbonalievna Khusanova
Lecturer, Fergana branch of TUIT named after Muhammad Al-Khorazmiy
ABSTRACT
In the section article, discrete convolution, general descriptions of the correlation process described in the correlation process and its effects on the impulse response, the results obtained in the Matlab program are explained in the example of practical results.
Keywords: Discrete convolution, correlation, impulse response, continuous signal, discrete signal, filter coefficients, sequences.
INTRODUCTION
Like a number of processes, convolution can be both permanent and discrete. Definitional convolution can be used for two functions or sequences, and control the outcome function or sequence at the output.
r-\ | [\
hi ^r—1 I
Figure 1. Mutual convolution of two functions
Constant convolution is a type of integral substitution, which is essentially very close to the correlation function.
y = f * g
Figure 2. Mutual convolution of two functions
Recalling the correlation function, we evaluated the degree of similarity of one function to another over time, in other words, we could calculate the area of change from one function to another and the area of overlap. The degree of similarity is reflected by the reflection of one function in the convolution, and its time is shifted by the second function. Let us now consider an example of the calculation of a discrete convolution, that is, the convolution of two sequences, knowing that in the transition from continuous signals to discrete, the integration operations are replaced by some
simple multiplications and sums. Consider two separate sequences of 5 samples each: x with values 4 2 1 2 3 and y with values 1 2 4 2 3.
Figure 3
To show the time shift, we fill the first sequence with zeros, reflect the second sequence, and multiply the number of elements in the element by the first five samples. Combining the elements of the multiplication results, we get the number 4. We then move the first sequence to a single report and repeat the multiplication and addition operations. Thus, we calculate the elements of the y sequence. As you can see, we moved the first x and rotated the second h sequence. Indeed, there is no fundamental difference in the significance of this relative shift, but such a demonstration is not in vain.
LITERATURE REVIEW
The first sequence can be thought of as an input signal, and the second as a type of system through which this signal passes. The result of the calculated linear discrete convolution includes more samples than each of the collected and the sequence. The built-in conv function can be used to quickly calculate the discrete convolution of MATLAB. In the command window, we create these two sequences and calculate their folds in one operation. Let's create the vectors x, h and calculate the resulting y vector. we visualize the output sequence using the stem function. (Figure 4)
» X = [4 2 1 2 3]
X = 20
4 2 12 3 t5
» h - [1 2 4 2 3] 10
h =
S
12 4 2 3
o-'-----
i 2 3 4 S 6 ? B 9
» y = conv(x,h)
y -
4 10 21 20 27 22 19 12 9
» stem(y)
A» _ _
Figure-4
As we can see, the result is exactly the same as previously calculated. We now move from abstract mathematical calculations to the practical applications of convolution for digital signal processing, but for this we need to return for a moment to the analog world. Let us recall the basics of combining RC chains, the theory of electrons.
Let's say we turn on the voltage source for a while and then turn it off. In the time range, the voltage dependence of the source is rectangular. We also describe its impulse response, i.e., the reaction of the chain to the delta function. Recall the Delta function or Dirac function, this function is not zero and is only zero at the beginning of the coordinates or at time t, i.e. an infinitely short infinitely strong pulse. Thus the reaction of the system to such impulses is called the impulse reaction. With its help we can describe the dynamics of the system.
DISCUSSION
In our case, to calculate the output signal, we need to combine the voltage across the capacitor, the input signal of the circuit voltage, with the pulsed reaction of the electron. Convolution is used to calculate the output of a system with a known input signal and a known characteristic of the system. Now, let's look at an example of using convolution to find signals. If the system has a pulsed response that is opposite to the value of the input sequence, then the result of the convolution operation will be maximal when the samples match. Based on this, it is possible to obtain the principle of compatible filtering. Appropriate filters are used when it is not important for us to maintain the received signal form, but when it is important to determine its presence on air and the time of arrival.
Figure-5
This is one of the usual radar functions, that is, if we know in advance the form of the signal we want to receive. It is not difficult to find the filter coefficients, i.e. the value of the second sequence for the convolution operation, otherwise they are equal to the signal report. In this case, the value at the filter output will be maximum when a noisy signal arrives at the input of the system. Similarly, signals can be detected in the background of noise.
Let's look at an example in MATLAB. In this example, we look for a random sequence of x using a filter that matches the pulse reaction of h. And we form the h impulse response by reflecting the sequence x. The fliplr function displays array elements.
RESULTS
We then add random elements before and after our sequence, as well as additional noise. We know where we put our sequence, so we write the last state of our sequence in the EXPECTED_NUM variable. We then calculate the conversion of our noisy signal
from the h pulse response, and the maximum value of the convolution should indicate the state of the end of our sequence. Let's check.
Figure-6
The graph above shows the noise signal and the actual sequences in the background, while the graph below shows our convolution. As you can see, it has a clear maximum.
Figure-7
CONCLUSION
We try to press the RUN button a few more times, i.e. the nature of the noise does not affect the state of the desired sequence. But if we replace the sequence with a few patterns, we can see in the same way that our convolution function successfully tracks its position against a background of noise.
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