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Communication Systems PHASOR & LINE SPECTRA By Engr. Dr. Jawwad Ahmad 1

Today’s Goal

 Phasor & Line Spectra

 One-Sided or Positive-Frequency Line Spectra  Conventions Regarding Line Spectra  Two-Sided or Double-Sided Line Spectra  Drill Problems

Engr. Dr. Jawwad Ahmad

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PHASOR & LINE SPECTRA  Consider the sinusoidal or ac (alternating-current) waveform v(t) as shown.

Engr. Dr. Jawwad Ahmad

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PHASOR & LINE SPECTRA  By convention, we express sinusoids in terms of the cosine function and write, => v(t )  A cos 

v(t )  A cos  t   

O => where, A is the peak value or amplitude and ω0 is the radian frequency.

 The phase angle  represents the fact that the peak has been shifted away from the time origin and occurs at t    . O  The reciprocal of the period equals the cyclical frequency fo, measured in cycles per seconds or hertz. 1 O fO   TO 2 Engr. Dr. Jawwad Ahmad

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PHASOR & LINE SPECTRA  The Phasor representation of a sinusoidal signal comes from Euler’s Theorem.  j e  cos   j sin  => Ree  j&  cos  Ime  j    j sin  where,  If we let ,   O t   e  j ( t  )  cos( O t   )  j sin( O t   ) => So, v' (t )  A cos( O t   )  j sin( O t   ) => O

 We can write any sinusoid as the real part of a complex exponential, namely; A cos  O t     A Re e j   t   Re v' (t ) 

v(t )



A cos O t   

Engr. Dr. Jawwad Ahmad







O



Re Ae j e jOt 5



PHASOR & LINE SPECTRA  This is called Phasor Representation because the term inside the brackets may be viewed as a rotating vector in a complex plane whose axes are the real and imaginary parts.

Engr. Dr. Jawwad Ahmad

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PHASOR & LINE SPECTRA  The Phasor has length A, rotates counter clockwise at the rate of revolutions per second, and at time makes an angle with respect to the positive real axis.  To describe the same Phasor in the frequency domain, associate the corresponding amplitude and phase with the particular frequency. Hence, a same frequency-domain description would be the line spectrum.  This is called one-sided or positive-frequency line spectra can be constructed for any linear combination of sinusoids.

Engr. Dr. Jawwad Ahmad

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PHASOR & LINE SPECTRA

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Four Conventions Regarding Line Spectra  These should be stated as; 1. In all our spectral drawings the independent variable will be cyclical frequency f hertz, rather than radian frequency ω, a shorthand notation for 2πf. 2. Phase angles will be measured with respect to cosine waves or equivalently,

sin  t  cos   t  90

3. Amplitude as always being a positive quantity. When negative sign appears, they must be absorb in the phase using +180 or –180  A cos  t  A cos   O t    4. Phase angles should be expressed in degrees. Engr. Dr. Jawwad Ahmad

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Four Conventions Regarding Line Spectra  To illustrate this consider the following signal,

=> w(t)

= 7 – 10 cos (40πt – 60o) + 4 sin 120 πt

Engr. Dr. Jawwad Ahmad

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Four Conventions Regarding Line Spectra  To illustrate this consider the following signal, => w(t) = 7 – 10 cos (40πt – 60o) + 4 sin 120 πt Using 1st convention: => w(t) = 7 – 10 cos (2π20t – 60o) + 4 sin (2π60t) Using 2nd convention: => w(t) = 7 cos(2π0t) – 10 cos(2π20t – 60o) + 4 cos(2π60t – 90o) Using 3rd convention: => w(t) = 7 cos(2π0t) + 10 cos(2π20t + 120o) + 4 cos(2π60t – 90o)

Engr. Dr. Jawwad Ahmad

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Four Conventions Regarding Line Spectra  To illustrate this consider the following signal, => w(t) = 7 cos(2π0t) + 10 cos(2π20t + 120o) + 4 cos(2π60t – 90o)

Engr. Dr. Jawwad Ahmad

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Two Sided Or Double Sided Frequency Line Spectra  But another spectral representation turns out to be more valuable, even though it involves negative frequencies.





1  Recalling that Re z   z  z * 2 Let => Z  A e j e jO t Then, => Z  A e  j e  jO t Therefore, 1 => Re Z   Z  Z *

 2

=> =>





1 A e j e j O t  A e  j e  j O t 2 A j jOt A  j  jOt Re Z   e e  e e 2 2

Re Z  

Engr. Dr. Jawwad Ahmad

 13

Two Sided Or Double Sided Frequency Line Spectra  So we now have a pair of conjugate Phasor. A A Re Ae j e jOt  A cos  O t     e j e jOt  e  j e  jOt 2 2





Conjugate Phasors

Engr. Dr. Jawwad Ahmad

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Two Sided Or Double Sided Frequency Line Spectra  Two Sided Spectrum

Engr. Dr. Jawwad Ahmad

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Two Sided Or Double Sided Frequency Line Spectra  This is the Double Sided Representation of the previous question.

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Two Sided Or Double Sided Frequency Line Spectra  CONCLUSION 

The line spectrum is two sided since it must include negative frequencies to allow for the opposite rotational directions, and one-half of the original amplitude is associated with each of the two frequencies  .f OThe amplitude spectrum has even symmetry while the phase spectrum has odd symmetry.



It should be emphasized that these line spectra, one-sided or two-sided, are just pictorial ways of representing sinusoidal of Phasor time function. A single line in the one-sided spectrum represents a real cosine wave; whereas a single line in the two-sided spectrum represents a complex exponential and the conjugate term must be add to get a real cosine wave.

Engr. Dr. Jawwad Ahmad

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Two Sided Or Double Sided Frequency Line Spectra  CONCLUSION 

Thus, speak of some frequency interval such as f1 to f2 in a two-sided spectrum also include the corresponding negative frequency interval –f1 to –f2. A simple notation for specifying both intervals is

f1  f  f2 

Finally, the amplitude spectrum displays the signals’ frequency contents.

Engr. Dr. Jawwad Ahmad

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Two Sided Or Double Sided Frequency Line Spectra  PRACTICE PROBLEM 

Develop One Sided and Two Sided Spectrum of the following: => v(t) =

–3 – 4 sin 30πt

Engr. Dr. Jawwad Ahmad

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Thank you

Engr. Dr. Jawwad Ahmad

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