Periodicity Professor Deepa Kundur Recall if a signal x(t) is periodic, then there exists a T > 0 such that x(t) = x(t + T ) University of Toronto If no T > 0 can be found, then x(t) is non-periodic. Professor Deepa Kundur (University of Toronto) 1 / 23 Professor Deepa Kundur (University of Toronto) 2 / 23 Continuous-time Sinusoids To find the period T > 0 of a general continuous-time sinusoid x(t) = A cos(ωt + φ): x(t) = x(t + T ) A cos(ωt + φ) = A cos(ω(t + T ) + φ) A cos(ωt + φ + 2πk) = A cos(ωt + φ + ωt ) 2πk = ωt T 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. Discrete-time Sinusoids To find the integer period N > 0 (i.e., (N Z + ) of a general discrete-time sinusoid x[n] = A cos(n + φ): where k Z. x[n] = x[n + N] A cos(n + φ) = A cos((n + N) + φ) A cos(n + φ + 2πk) = A cos(n + φ + N) 2πk = N Note: there may not exist a k Z such that 2πk is an integer. Professor Deepa Kundur (University of Toronto) 3 / 23 Professor Deepa Kundur (University of Toronto) 4 / 23
Discrete-time Sinusoids Example i: = 37 11 π Example ii: = 2 37 11 π = 22 37 k N 0 = 22 k = 22 for k = 37; x[n] is periodic. 37 N = 2πk 2 = πk N Z + does not exist for any k Z; x[n] is non-periodic. Example iii: = 2π N = 2πk 2π = 2k N Z + does not exist for any k Z; x[n] is not periodic. Professor Deepa Kundur (University of Toronto) 5 / 23 Discrete-time Sinusoids N = 2π k N = π 2k N }{{} RATIONAL Therefore, a discrete-time sinusoid is periodic if its radian frequency is a rational multiple of π. Otherwise, the discrete-time sinusoid is non-periodic. Professor Deepa Kundur (University of Toronto) / 23 Example 1: = π/ = π 1 x[n] = cos ( πn ) π 1 = 12k N 0 = 12 for k = 1 The fundamental period is 12 which corresponds to k = 1 envelope cycles. ENVELOPE CYCLES Professor Deepa Kundur (University of Toronto) 7 / 23 Professor Deepa Kundur (University of Toronto) 8 / 23
Example 2: = 8π/31 = π 8 31 x[n] = cos ( ) 8πn 31 π 8 31 N 0 = 31 for k = 4 = 31 4 k The fundamental period is 31 which corresponds to k = 4 envelope cycles. ENVELOPE CYCLES Professor Deepa Kundur (University of Toronto) 9 / 23 Professor Deepa Kundur (University of Toronto) 10 / 23 Example 3: = 1/ = π 1 π ( n x[n] = cos ) N = 2πk 1 = 12πk N Z + does not exist for any k Z; x[n] is non-periodic. NOT PERIODIC Professor Deepa Kundur (University of Toronto) 11 / 23 Professor Deepa Kundur (University of Toronto) 12 / 23
Continuous-Time Sinusoids: Frequency and Rate of ω smaller x(t) = A cos(ωt + φ) T = 2π ω = 1 f Rate of oscillation increases as ω increases (or T decreases). Professor Deepa Kundur (University of Toronto) 13 / 23 Professor Deepa Kundur (University of Toronto) 14 / 23 ω larger, rate of oscillation higher Continuous-Time Sinusoids: Frequency and Rate of Also, note that x 1 (t) x 2 (t) for all t for when ω 1 ω 2. x 1 (t) = A cos(ω 1 t + φ) and x 2 (t) = A cos(ω 2 t + φ) Professor Deepa Kundur (University of Toronto) 15 / 23 Professor Deepa Kundur (University of Toronto) 1 / 23
: Frequency and Rate of MINIMUM OSCILLATION x[n] = A cos(n + φ) Rate of oscillation increases as increases UP TO A POINT then decreases again and then increases again and then decreases again.... MAXIMUM OSCILLATION MINIMUM OSCILLATION Professor Deepa Kundur (University of Toronto) 17 / 23 Professor Deepa Kundur (University of Toronto) 18 / 23 : Frequency and Rate of : Frequency and Rate of Let x[n] = A cos(n + φ) x 1 [n] = A cos( 1 n + φ) and x 2 [n] = A cos( 2 n + φ) and 2 = 1 + 2πk where k Z: Discrete-time sinusoids repeat as increases! x 2 [n] = A cos( 2 n + φ) = A cos(( 1 + 2πk)n + φ) = A cos( 1 n + 2πkn + φ) = A cos( 1 n + φ) = x 1 [n] Professor Deepa Kundur (University of Toronto) 19 / 23 Professor Deepa Kundur (University of Toronto) 20 / 23
Professor Deepa Kundur (University of Toronto) 21 / 23 Professor Deepa Kundur (University of Toronto) 22 / 23 Discete-Time Sinusoids: Frequency and Rate of x[n] = A cos(n + φ) can be considered a sampled version of x(t) = A cos(t + φ) at integer time instants. As increases, the samples miss the faster oscillatory behavior. Professor Deepa Kundur (University of Toronto) 23 / 23