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DstTimeSinusoids_handouts - Discrete-Time Sinusoids...

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Discrete-Time Sinusoids Dr. Deepa Kundur Texas A&M University Dr. Deepa Kundur (Texas A&M University) Discrete-Time Sinusoids 1 / 23 Discrete-Time Sinusoids Periodicity Recall if a signal x ( t ) is periodic , then there exists a T > 0 such that x ( t ) = x ( t + T ) If no T > 0 can be found, then x ( t ) is non-periodic . Dr. Deepa Kundur (Texas A&M University) Discrete-Time Sinusoids 2 / 23 Discrete-Time Sinusoids Continuous-time Sinusoids To find the period T > 0 of a general continuous-time sinusoid x ( t ) = A cos( ω t + φ ) : x ( t ) = x ( t + T ) A cos( ω t + φ ) = A cos( ω ( t + T ) + φ ) A cos( ω t + φ + 2 π k ) = A cos( ω t + φ + ω T ) 2 π k = ω T T = 2 π k ω where k Z . Note : when k is the same sign as ω , T > 0 . Therefore, there exists a T > 0 such that x ( t ) = x ( t + T ) and therefore x ( t ) is periodic . Dr. Deepa Kundur (Texas A&M University) Discrete-Time Sinusoids 3 / 23 Discrete-Time Sinusoids Discrete-time Sinusoids To find the integer period N > 0 (i.e., ( N Z + ) of a general discrete-time sinusoid x [ n ] = A cos(Ω n + φ ) : x [ n ] = x [ n + N ] A cos(Ω n + φ ) = A cos(Ω ( n + N ) + φ ) A cos(Ω n + φ + 2 π k ) = A cos(Ω n + φ + Ω N ) 2 π k = Ω N N = 2 π k Ω where k Z . Note : there may not exist a k Z such that 2 π k Ω is an integer. Dr. Deepa Kundur (Texas A&M University) Discrete-Time Sinusoids 4 / 23
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Discrete-Time Sinusoids Discrete-time Sinusoids Example i : Ω = 37 11 π N = 2 π k Ω = 2 π k 37 11 π = 22 37 k N 0 = 22 37 k = 22 for k = 37; x [ n ] is periodic.
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