Section 3 Discrete-Time Signals EO 2402 Summer 2013 07/05/2013 EO2402.SuFY13/MPF Section 3 1
[p. 3] Discrete-Time Signal Description Sampling, sampling theorem Discrete sinusoidal signal Discrete exponential signal Unit impulse function Unit sequence function Signum function Unit ramp signal Periodic impulse function Even/Odd Products of Even and Odd Signals [p. 21] Symmetric Finite Summation [p. 22] Periodic Signal [p. 26] Energy & Power Signals 07/05/2013 EO2402.SuFY13/MPF Section 3 2
Discrete-Time Signal Description - Sampling - A discrete signal may be obtained by sampling a continuous signal - Sampling refers to acquiring values of a signal at discrete points in time xn [ ] xn ( ) xnt ( ), where T is the time s between samples s 07/05/2013 EO2402.SuFY13/MPF Section 3 3
Discrete Sinusoidal Signal xn [ ] Acos(2 f n ) Question: Is x(n) necessarily periodic? 0 07/05/2013 EO2402.SuFY13/MPF Section 3 4
Is x[n] a periodic signal? 07/05/2013 EO2402.SuFY13/MPF Section 3 5
Question: Is there a unique association between an analog signal and its discrete derivation? 07/05/2013 EO2402.SuFY13/MPF Section 3 6
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Sampling Theorem Sampling Theorem: A unique association between an analog signal and its discrete derivation is guaranteed provided that the sampling rate f s used to sample the continuous-time signal x(t) is equal to at least twice the maximum frequency f max contained in the signal x(t) Nyquist rate: Sampling rate when selected as f s =2f max Definitions: When f s >2f max, signal is said to be oversampled When f s <2f max, signal is said to be undersampled (leads to aliasing) 07/05/2013 EO2402.SuFY13/MPF Section 3 8
Discrete Exponential Signal Real n x[ n] A Ae n Complex Real/imaginary plots 07/05/2013 EO2402.SuFY13/MPF Section 3 9
Discrete Exponential Signal, cont Real n x[ n] A Ae n Complex Magnitude/Phase plots 07/05/2013 EO2402.SuFY13/MPF Section 3 10
The Unit Impulse Function n 1, n 0 0, n 0 [n] is defined as the discrete-time unit impulse ( Kronecker delta function) [n] has a sampling property, n x x A n n n A n 0 0 [n] does NOT have a scaling property. n an, a 0. 07/05/2013 EO2402.SuFY13/MPF Section 3 11
Unit Sequence Function u n 1, n 0 0, n 0 07/05/2013 EO2402.SuFY13/MPF Section 3 12
The Signum Function 1, n 0 sgn n 0, n 0 1, n 0 07/05/2013 EO2402.SuFY13/MPF Section 3 13
Unit Ramp Function ramp n n, n 0 0, n 0 nu n n m um 1 07/05/2013 EO2402.SuFY13/MPF Section 3 14
Periodic Impulse Function N n m n mn 07/05/2013 EO2402.SuFY13/MPF Section 3 15
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Even / Odd Signals - Definition: A Signal x[n] is odd if x[n]=-x[-n] A Signal x[n] is even if x[n]=x[-n] Property: Any signal x[n] can be decomposed into a sum of even and odd components x[ n ] xe [ n] x o [ n ] 0.5 x[ n] x[ n] 0.5 x[ n] x[ n] 07/05/2013 EO2402.SuFY13/MPF Section 3 17
Products of Even and Odd Functions Two Even Functions 07/05/2013 EO2402.SuFY13/MPF Section 3 18
Products of Even and Odd Function, cont An Even Function and an Odd Function 07/05/2013 EO2402.SuFY13/MPF Section 3 19
Products of Even and Odd Functions, cont Two Odd Functions 07/05/2013 EO2402.SuFY13/MPF Section 3 20
Symmetric Finite summation (Discrete equivalent of integration) Even function Odd function N N xn [ ] x[0] 2 xn [ ] xn [ ] 0 kn k1 N kn 07/05/2013 EO2402.SuFY13/MPF Section 3 21
Periodic Signal A discrete signal xn [ ] is periodic iff 0 xn [ ] xn [ mn], where N and m are integer N 0 is called the Period 0 07/05/2013 EO2402.SuFY13/MPF Section 3 22
Example: Find the period of the signal x[n]=cos(2n/36)+sin(10n/24) y[n]=cos(5n/13)+sin(8n/39) 07/05/2013 EO2402.SuFY13/MPF Section 3 23
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Energy / Power Signals - Definition: The energy in a signal x[n] over the interval N=[N 1,N 2 ] is defined as E N N 2 k N 1 x[ k ] 2 - Definition: The energy in a signal x[n] is defined as: E x k x[ k ] - Definition: A signal with finite energy is called an energy signal 2 07/05/2013 EO2402.SuFY13/MPF Section 3 26
Signal Energy/Power, cont Find the signal energy of x n N N 1 n Note: r 1 r n0 1 r N 2 n 5 / 3, 0 n 8 0, otherwise, r 1, r 1 07/05/2013 EO2402.SuFY13/MPF Section 3 27
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Energy / Power Signals, cont - Definition: The average power of a signal x[n] over the interval N=[N 1,N 2 ] is defined as P N 1 2 1 k N N N 2 N 1 x[ k ] 2 - Definition: The average power of a signal x[n] is defined as: P x lim N 1 2 N k N 1 N x k - Definition: For a periodic signal, the average power of x[n] is defined as: 1 2 P x N k N x[ k ] - Definition: A signal with finite power is called a power signal 2 07/05/2013 EO2402.SuFY13/MPF Section 3 29
Signal Energy/Power, cont Find the average signal power of n x 2sgn[ n] 4 07/05/2013 EO2402.SuFY13/MPF Section 3 30
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