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ece642-1-3 - ECE 642 COMMUNICATION SYSTEMS I Lecture 1 Part...

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1 I. Review of Time-Frequency Relations © 2002 by Jacob Klapper ECE 642 - COMMUNICATION SYSTEMS I Lecture 1, Part 3 Review of Time-Frequency Relations Professor Jacob Klapper Department of Electrical and Computer Engineering New Jersey Institute of Technology I. Review of Time-Frequency Relations © 2002 by Jacob Klapper Fourier Series A periodic x(t) can be written as a possibly infinite sum of harmonic phasors. (1-1) where C n = complex constant depending on n 0 jn ω t x(t)= C e n n=- I. Review of Time-Frequency Relations © 2002 by Jacob Klapper Fourier Series (cont’d) Equivalently, 0 0 n 1 0 n 1 x(t)=A A cosn t n B sinn t n = = + ω + ω (1-2) where A 0 , A n , B n = real constants
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2 I. Review of Time-Frequency Relations © 2002 by Jacob Klapper Exercise Can you sketch the frequency domain from (1-1) or (1-2) ? What provides the amplitude and phase ? One –sided or two-sided spectra ? Note: Frequency domain is discrete in f I. Review of Time-Frequency Relations © 2002 by Jacob Klapper Example 6 3 6 6 3 3 1.5 1.5 2 50 2 100 2 50 2 50 2 100 2 100 2 3 t t t t t t x(t) cos( ) cos( ) j( ) j( ) e e j( ) j( ) e e π π π π π π π⋅
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