Signals and Systems Lecture 20: Chapter 4 Sampling & Aliasing

2 Today's lecture  Spectrum for Discrete Time Domain –Oversampling –Under=sampling  Sampling Theorem  Aliasing  Ideal Reconstruction  Folding

3 General Formula for Frequency Aliases  Adding any integer multiple of 2π gives an alias = 0.4 π + 2 πl l = 0,1,2,3,…..  Another alias x 3 [n] = cos(1.6πn) x 3 [n] = cos(2πn - 0.4πn) x 3 [n] = cos(0.4πn) Since cos (2πn - θ) = cos (θ )  All aliases maybe obtained as, + 2 πl, 2 πl - l = integer ooo

4 Folded Aliases  Aliases of a negative frequency are called folded aliases Acos (2πn - n - θ) = Acos ((2π - )n- θ) = Acos (- n- θ) = Acos ( n + θ)  The algebraic sign of the phase angles of the folded aliases must be opposite to the sign of the phase angle of the principal alias. o oo o

5 Spectrum of a Discrete-Time Signal y 1 [n] = 2cos(0.4πn)+ cos(0.6πn) y 2 [n] = 2cos(0.4πn)+ cos(2.6πn)

6 Spectrum for x[n]

7 Digital Frequency and Frequency Spectrum

8 Spectrum of a Discrete-time signal obtained by sampling  Starting with a continuous-time signal x[t] = A cos(ω o t +  )  Spectrum consists of two spectral lines at +ω o with complex amplitudes 1/2A e +jφ  The sampled discrete-time signal x[n] = A cos((ω o / f s )n+  ) x[n] = 1/2Ae +jφ e + j(ωo/ fs )n + 1/2Ae - jφ e - j(ωo/ fs )n  Has two spectrum lines at ώ = +ω o / f s, but it also must contain all the aliases at the following discrete-time frequencies ώ = + ω o / f s + 2πll=0, +1, +2,… ώ = - ω o / f s + 2πl l=0, +1, +2,…

9 Spectrum (Digital) with Over-sampling

10 Spectrum (Digital) with fs = f (under-sampling)

11 Sampling Theorem  Continuous-time signal x(t) with frequencies no higher than f max can be reconstructed from its samples x(k T s ) if samples taken at rate f s > 2 f max Nyquist rate = 2 f max Nyquist frequency = f s / 2 –Sampling theorem also suggests that there should be two samples per cycle.  Example: Sampling audio signals Normal human ear can hear up to 20 kHz

12 Sampling Theorem