Effect of Convolution between Two Complex Waves having the ...

Int. J. of Mathematical Sciences and Applications, Vol. 1, No. 3, September 2011 Copyright  Mind Reader Publications www.journalshub.com

Effect of Convolution between Two Complex Waves having the same angular frequencies with Amplitude Spectrum Analysis in Frequency Domain A.N.M. Rezaul Karim Department of Computer Science & Engineering International Islamic University Chittagong, Bangladesh E-mail: [email protected] Abstract As digital signal processing continues to emerge as a major discipline in the field of electrical engineering, an even greater demand has evolved to understand the basic theoretical concepts involved in the development of varied and diverse signal processing systems. The most fundamental concepts employed are the sampling theorem, Fourier transforms, convolution, covariance, etc. The intent of this paper will be to address the concept of convolution and to present Convolved signal between Two Complex Waves having the same angular frequencies with Amplitude Spectrum Analysis in Frequency Domain The Fourier series gives us a very important and useful way of representing an image while it is a very powerful representation for many reasons, we will mainly introduce it as a way of understanding the effect of Convolution between two complex waves. Key Words: Convolution, Fourier series, Frequency Spectrum, Attenuation 1. Introduction Perhaps the easiest way to understand the concept of convolution would be an approach that initially clarifies a subject relating to the frequency spectrum of linear networks. Determining the frequency spectrum or frequency transfer function of a linear network provides one with the knowledge of how a network will respond to or alter an input signal. Conventional methods used to determine this entail the use of spectrum analyzers which use either sweep generators or variable-frequency oscillators to impress upon a network all possible frequencies of equal amplitude and equal phase. The response of a network to all frequencies can thus be determined. Any amplitude and phase variations at the output of a network are due to the network itself and as a result define the frequency transfer function. Another means of obtaining this same information would be to apply an impulse function to the input of a network and then analyze the network impulse-response for its spectral frequency content. Comparison of the network-frequency transfer function obtained by the two techniques would yield the same information. The characteristics of electrical (and other) signals can be explored in two ways. The first is to examine their waveforms in the time domain, which can be done using an oscilloscope. The second is to examine their spectra in the frequency domain, which can be done using a spectrum analyzer. Both descriptions are useful, although frequency domain analysis is the most natural tool for many tasks in electrical circuit design, speech and music analysis, and for video signals. In fact, frequency domain analysis is of fundamental importance in all branches of engineering, from the design of bridges and roads to wireless Communications. Many Signals are more convenient to process, analyze, Synthesize and/or compress in the Frequency Domain 1.1 A signal can be described in two spaces: time domain and frequency domain. The techniques of signals analysis both in time domain and frequency domain are called methods of time-frequency analysis [1, 2]. 1.2 Fourier series (FS) [3] is an ideal tool of spectrum analysis for stationary signals for which the spectrum is not a function of the time. The power of Fourier series is that it allows the decomposition of a signal into individual frequency components.

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Figure: 01: sine wave shifted in phase by 1.10.2 Angular Speed or Angular Frequency =;

90 0

2 t

Figure: 02: Angular Frequency

  2    t    t 1.10.3 We have, a cos  b sin   R sin(  ) The above relationships are obtained from the trigonometric identity Where, Phase   tan

1

a   and R  a 2  b 2 b

1.10.4. Mathematical Notation Fourier series .Mathematically, a periodic signal f (t) may be represented by a Fourier series as follows

f (t )  A0  A1 sin(t  1 )  A2 sin(2t   2 )  .......... ......  An sin(nt   n )  .......... ......( 1) The term A1 sin(t  1 ) is called the first harmonic or the fundamental mode, and it has the same

 as the parent function f (t ) . The term An sin(nt   n ) is called the nth harmonic and it has frequency n , which is n times that of the fundamental. An denotes the amplitude of the frequency

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nth harmonic and  n is its phase angle, measuring the lag or lead of the nth harmonic with reference to a pure sine wave of the same frequency[8] Since,

A n sin(nt   n )  A n [sin nt cos  n  cos nt sin  n ] A n sin(nt   n )  ( A n cos  n ) sin nt  ( A n sin  n ) cos nt A n sin(nt   n )  ( A n sin  n ) cos nt  ( A n cos  n ) sin nt A n sin(nt   n )  a n osnt  b n sin nt Where,

a n  A n sin  n and b n  A n cos  n The expansion (1) may be written as f (t ) 

  a0  a n cos(n 0 t )  b n sin(n 0 t ) ---------------------------(2) 2 n 1 n 1





Where the variables are: a0 1 The average (DC) value of the signal: a 0  2 



 f (t)dt



n The harmonic number: 1=fundamental, 2=2nd harmonic, etc a n Peak value of the magnitude of the nth cosine harmonic

b n Peak value of the magnitude of the nth sine harmonic  0 Fundamental frequency,  0 

2 T

T Period of f (t ) The cosine and sine waves occur at multiples of the fundamental angular frequency 1 2 0  = 2f 0 [ f 0  ] . T T The terms a n and b n determine the magnitude of each harmonic. These are the quantities that show up as vertical lines on a spectrum plot

Where, amplitude Cn

a  2 2  a n  b n and phase  n  tan 1  n   bn 

2. Materials and Methods Convolution is a mathematical operator which computes the point wise overlap between two functions Note that convolution in the time domain is equivalent to multiplication in the frequency domain (and vice versa) Convolution is basic operations that we will perform to extract information from images. Convolving two waveforms in the time domain means that we are multiplying their spectra (i.e. frequency content) in the frequency domain. By "multiplying" the spectra we mean that any frequency that is strong in both signals will be very strong in the convolved signal, and conversely any frequency that is weak in either input signal will be weak in the output signal [9] In practice, a relatively simple application of convolution is where we have the "impulse response" of a space However, the output duration of a convolved signal is the sum of the durations of the two inputs. Some basic principles and the means for performing the analysis with MATLAB have been considered in this paper 3. Test and results 3.1 Let a periodic function g (t ) of period

2 is defined by

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Effect of Convolution between Two Complex Waves…

 3 ;    t  0 g(t )    3 ;0  t   g(t  2)  g(t) 6  1 The function g (t )   (1  cos n ) sin nt represent a Fourier series for the above function  n 1 n a0 2 an

0

0 6 (1  cosn) bn  n 6 (1  cosn) n

 cn  an  bn  0  bn  bn  2

2

2

  t  nt    n that angular frequency are same of cosine and sine signal It has frequency components at frequencies

 n  n  n.1  n 0 (n  1,2,3,......... .......... ......) Where  0 is the frequency of the parent function g (t ) The amplitude C n is drawn with respect to angular frequency   n . 4 3 2

g(t)

1 0 -1 -2 -3 -4 -8

-6

-4

-2

0 time domain

2

4

6

8

Figure 03: A complex wave g (t) 4 3.5 3

amplitude

2.5 2 1.5 1 0.5 0

1

2

3

4

5 6 frequency domain

7

8

Figure: 04: Amplitude Spectrum of g (t )

3.2 Again, Let a periodic function h(t ) of period

2 is defined by

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A.N.M. Rezaul Karim

 t ;    t  0 h(t )    t ; 0t h(t  2)  h(t)  2  1 The function h(t )    2 (cos n  1) cos nt represent a Fourier series for the above 2  n 1 n function  a0  2 2

2 (cos n  1)  n2 bn  0 an 

2 2 2  cn  an  bn  an  0  an 

2 (cos n  1)  n2

  t  nt    n that angular frequency are same of cosine and sine signal It has frequency components at frequencies

 n  n  n.1  n 0 (n  1,2,3,......... .......... ......) Where  0 is the frequency of the parent function h(t ) The amplitude C n is drawn with respect to angular frequency   n . 3.5

3

2.5

h(t)

2

1.5

1

0.5

0 -8

-6

-4

-2

0

2 4 time domain

6

8

10

12

Figure 05: A complex wave h (t) 0

-0.2

amplitude

-0.4

-0.6

-0.8

-1

-1.2

-1.4

1

2

3

4

5 6 frequency domain

7

8

9

Figure: 06: Amplitude Spectrum of h(t )

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Effect of Convolution between Two Complex Waves…

3.3. Convolution convolved signal: g(t)*h(t) 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5

2

4

6

8

10

12

14

16

18

20

Figure: 07: Convolved Signal 4. Findings and Discussion Conventional methods used to determine this entail the use of spectrum analyzers which use either sweep generators or variable-frequency oscillators to impress upon a network all possible frequencies of equal amplitude and equal phase. In fact, one can convolve any sound with another, not just an impulse response. In that case, we are "filtering" the first sound through the spectrum of the second, such that any frequencies the two sounds have in common will be emphasized. A particular case is where we convolve the sound with itself, thereby guaranteeing maximum correlation between the two sources. In this case, prominent frequencies will be exaggerated and frequencies with little energy will be attenuated. This example showed how the response could be determined using the convolution. The spectrum after convolution of two complex waves shows the losing local information of the spectrum 5. Conclusion This article attempted to implement the convolution between two complex waves in frequency domain .These examples showed how the response occurred of amplitude spectrum of complex waves by convolution. The spectrum after convolution of Fourier series (complex waves) shows the losing local information of the spectrum. To do research convolution between complex waves having different angular frequency is our future work. References [I] Leon Cohen,Time-Frequency Distributions-A Review. Proceeding of the IEEE, Vol. 77.No.7, July 1989, pp941-981. [2] Leon Cohen, Patrick Loughlin, recent development in time-frequency analysis, Kluwer academic Publishers, 1998 [3] Okan K. Ersoy, A comparative Review of Real and Complex Fourier- Related Transforms. Proceedings of the IEEE,vo1.82,no.3, march 1994,pp429-447 [4] GLYN JAMES, Advanced Modern Engineering Mathematics, 2 nd edition, pp 338 [5] GLYN JAMES, Advanced Modern Engineering Mathematics, 2nd edition, pp 281 [6] Audio Spectrum Analyse Peter Hiscocks, Wireless World Magazine, May 1980, pp55-60 An audiospectrum analyser that is suitable for construction by hobbyists. [7 ]http://en.wikipedia.org/wiki/Attenuation [8] GLYN JAMES, Advanced Modern Engineering Mathematics, 2 nd edition, pp 283 [9] Curtis Roads, the Computer Music Tutorial, MIT Press, 1996

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