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Ch 5 and 9 lecture

# Ch 5 and 9 lecture - EE 3120 Chapter 3 Time Domain Analysis...

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EE 3120 Chapter 3 Time Domain Analysis of Discrete-Time Systems Lecture 2

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N = 20 N = 20 sin 0.1 k π sin 0.3 k π Both sequences have the same fundamental period and thus the same frequency. Obviously one sequence changes more rapidly than the other. The way we define frequency should not be used.
We cannot use the fundamental period of a sequence to define its frequency. We must define it using analog sinusoids. An analog is called an envelope of if the sample of with sampling period T equals sin t ϖ sin o kT ϖ sin t ϖ sin o kT ϖ sin sin o t kT kT t ϖ ϖ = = for all k There are many envelopes of because for every integer n is also an envelope of sin o kT ϖ sin( 2 ) n T t ϖ π + sin o kT ϖ The primary envelope of is defined as the envelope with the smallest frequency in magnitude. Which is the analog with: sin o kT ϖ sin t ϖ T π ϖ π - < or T T π π ϖ - < 2 sin( ) sin( 2 ) sin cos( 2 ) cos sin( 2 ) sin t kT n t kT nk kT nk kT nk kT T π ϖ ϖ π ϖ π ϖ π ϖ = + = + = + = There are infinitely many envelopes. sin sin o t kT kT t ϖ ϖ = = Such that . Then the frequency if the sequence is defined as the frequency of the analog signal.

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sinπ k with T = 1 has envelopes Definition: The frequency of a digital sin o kT ϖ , periodic or not, is defined as the frequency of the analog sin ωt with ω whose sample, with sampling period T equals sin o kT ϖ sin3 sin( ) sin5 sin( 3 ) t t t t π π π π - - And any sin( 2 ) n t π π +
sin 7.1

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