Lesson_41 - EEL 3135: Signals and Systems Dr. Fred J....

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EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor Lesson Title: Spectral Representation Lesson Number: 4 (Sections 3.1-3.4) Spectrum = frequency domain representation of time-domain signal Introduction Consider the continuous time-domain signal x(t) shown below. ( 29 ( 29 = + + = N k k k k t A A t x 1 0 cos φ ϖ 1. The signal can be presented in a more elegantly using complex exponentials and Euler’s equation. Noting tha for ϖ k =2 π f k : ( 29 t f j t f j k k k e e t f π 2 2 2 1 2 1 2 cos - + = 2. it follows that: ( 29 k k k j k k N k t f j k t f j k e A X e X e X X t x = + + = = - ; 2 2 1 2 2 0 3. The single ended summation, shown in Equation 3, can now be replaced by the double ended summation, namely: ( 29 = = = - = 0 if , 5 . 0 0 if , ; 0 2 k e A k A a e a t x k k j k k N N k t f j k 4. where the coefficients a k are, in general, complex numbers. The two-sided spectrum, defined by the double-ended sum of Equation 4, can be graphically represented as: Magnitude and phase response (phasor) Real and imaginary spectrum (complex) which displays the frequency spectrum of the signal under study. Variations on this theme are: Log (dB) magnitude vs. linear frequency (semi-log) Log (dB) magnitude vs. log frequency (log-log or Bode plot) 1
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EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor Example Consider the cosine wave x(t)=cos( ϖ 0 t) = (1/2)e j ϖ 0 t + (1/2)e -j ϖ 0 t for some ϖ 0 . Display the magnitude and phase frequency response. 0.5 ------------------------------------------------------------------ 0 - ϖ 0 0 ϖ 0 0 ° - ϖ 0 0 ϖ 0 Display the real/imaginary frequency response 0.5 ------------------------------------------------------------------ 0 - ϖ 0 0 ϖ 0 0 - ϖ 0 0 ϖ 0 Figure 1: Spectral representation of a cosine wave. Consider now the sine wave x(t)=sin( ϖ 0 t) = (1/2j)e j ϖ 0 t - (1/2j)e -j ϖ 0 t = (-j/2)e j ϖ 0 t + (j/2)e -j ϖ 0 t for some ϖ 0 . Display the magnitude and phase frequency response. 0.5 ------------------------------------------------------------------ 0 - ϖ 0 0 ϖ 0 2 Magnitude Phase Real Imaginary Magnitude Imag. Real - ϖ 0 0 ϖ 0
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EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor 0 ° - ϖ 0 0 ϖ 0 The real/imaginary frequency response is 0 - ϖ 0 0 ϖ 0 0.5 ------------------------------------------------------------------ 0 - ϖ 0 0 ϖ 0 -0.5 ------------------------------------------------------------------ Figure 2: Spectral representation of a sine wave. Beat Frequency Sinusoids can be multiplied or added and the spectral outcomes predicted using simple trigonometry. Specifically, for multiplication (product modulation) opérations, where : x 1 (t)=cos( ϖ 1 t); x 2 (t)=cos( ϖ 2 t); 5. it follows that : y(t)= x 1 (t)*x 2 (t)= cos( ϖ 1 t)* cos( ϖ 2 t)= (1/2)cos( ϖ 1 + ϖ 2 )t) + (1/2)cos( ϖ 1 - ϖ 2 )t) 6. This results in what is called a sum and difference spectrum. Example Consider the case where x 1 (t)=cos( ϖ 1 t) = x 2 (t). Therefore y(t)= x 1 (t)*x 2 (t)= (1/2)cos(2 ϖ 1 t) + (1/2)cos(0t)=(1/2)+(1/2)cos(2 ϖ 1 t). To illustrate : 3 -90 ° 90 ° Phase Real Imaginary Imag. Real 0 ϖ 0
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EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor {-1,1}, ϖ 1 x 1 (t)=x 2 (t)= cos( ϖ 1 t) {0,1}, 2 ϖ 1 y(t)=x 1 (t)*x 2 (t)=(1/2)+(1/2)cos(2 ϖ 1 t). Figure 3 : Product spectrum For a summation, if : x 1 (t)=cos( ϖ 1 t); x 2 (t)=cos( ϖ 2 t); 7.
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Lesson_41 - EEL 3135: Signals and Systems Dr. Fred J....

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