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EE104: Lecture 7 Outline

EE104: Lecture 7 Outline. Review of Last Lecture Time Scaling Duality Frequency Shifting (Modulation) Multiplication in Time Convolution in Time. Key Fourier Transform Properties. Review of Last Lecture: Useful FT Properties. Linearity Saves work in computing new FTs Time Shift

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EE104: Lecture 7 Outline

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  1. EE104: Lecture 7 Outline • Review of Last Lecture • Time Scaling • Duality • Frequency Shifting (Modulation) • Multiplication in Time • Convolution in Time Key Fourier Transform Properties

  2. Review of Last Lecture:Useful FT Properties • Linearity • Saves work in computing new FTs • Time Shift • Leads to linear phase shift in frequency • Differentiation in Time • Not common in communications, used more in linear systems • Integration in Time • Common operation in digital demodulation • DC property: • Conjugation • Indicates symmetries in FT of real signals • Parseval’s Relation • Compute signal power in either time or frequency domains

  3. Key Properties of FTs • Time scaling • Contracting the time axis leads to an expansion of the frequency axis • Duality • Symmetry between time and frequency domains • “Reverse the pictures” • Eliminates half the transform pairs • Frequency shifting (modulation) • Modulation (multiplying a time signal by an exponential) leads to a frequency shift.

  4. X(f) X(f)H(f) H(f) More Key Properties • Multiplication in Time • Becomes complicated convolution in frequency • Mod/Demod often involves multiplication • Time windowing becomes frequency convolution w/sinc • Convolution in Time • Becomes multiplication in frequency • Defines output of LTI filters: easier to analyze with FTs sinc(f) X(f) * rect(t) x(t) x(t)*h(t) x(t) h(t)

  5. 2 2 z(-4-t) z(-6-t) z(1-t) z(-2-t) z(-1.99-t) z(0-t) z(2-t) z(-1-t) 2 1 .01 Convolution Review • y(t)=x(t)*z(t)= x(t)z(t-t)dt • Flip one signal and drag it across the other • Area under product at drag offset t is y(t). z(t) x(t) z(t-t) x(t) z(t) t t t t t t-1 t t+1 -1 0 1 -1 0 1 x(t) t 2 -6 -1 0 1 -4 -2 y(t) 2 -4 t 0 -2 -6 -1 0 1

  6. Main Points • Stretching a signal along the time axis causes it to shrink along the frequency axis. • Time and frequency domains are duals. • Frequency shifting obtained by modulation in time • Multiplication in time becomes convolution in frequency • Time limited signals cannot be bandlimited • Filter outputs obtain by convolution • Convolution is a drag (and a flip) • Easier to analyze in the frequency domain

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