# R03 - ENEE 241 02 READING ASSIGNMENT 3 Tue 02/10 Lecture...

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READING ASSIGNMENT 3 Tue 02/10 Lecture Topic: discrete-time sinusoids; sampling of continuous-time sinusoids Textbook References: sections 1.5, 1.6 Key Points: The discrete time parameter n counts samples. The (angular) frequency parameter ω is an angle increment (radians/sample). Physical time (seconds) is nowhere involved. Frequencies ω and ω + 2 π are equivalent (i.e., produce the same signal) for real or complex sinusoids in discrete time. Frequencies ω and 2 π - ω can be used alternatively to describe a real sinusoid in discrete time: cos( ωn + φ ) = cos( - ωn - φ ) = cos((2 π - ω ) n - φ ) The eﬀective range of frequencies for a real sinusoid in discrete time is 0 (lowest) to π (highest). A discrete-time sinusoid is periodic if and only if ω is of the form ω = k N · 2 π for integers k and N . The fundamental period is the smallest value of N for which the above holds. Sampling a continuous-time sinusoid at a rate of f s = 1 /T s (samples/second) produces a discrete-time sinusoid. If the continuous-time sinusoid has angular frequency Ω = 2 πf = 2 π/T , the resulting discrete- time sinusoid has angular frequency ω = Ω T s = 2 π · f f s = 2 π · T s T At high sampling rates, discrete-time samples capture the variation of the continuous-time signal in great detail. Two diﬀerent sampling rates f s = 1 /T s and f 0 s = 1 /T 0 s will produce samples having the same eﬀective frequency provided the sum T s + T 0 s or the diﬀerence T s - T 0 s is an integer multiple of T = 1 /f . Theory and Examples: 1 . A discrete-time signal is a sequence of values (samples) x [ n ], where n ranges over all integers. A discrete-time sinusoid has the general form x [ n ] = A cos( ωn + φ ) or, in its complex version, z [ n ] = Ae j ( ωn + φ ) Question: How is x [ n ] related to z [ n ]? 1

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R03 - ENEE 241 02 READING ASSIGNMENT 3 Tue 02/10 Lecture...

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