Download presentation
Presentation is loading. Please wait.
Published byJasmin Eustacia Williamson Modified over 9 years ago
1
Signals and Systems Lecture 20: Chapter 4 Sampling & Aliasing
2
2 Today's lecture Spectrum for Discrete Time Domain –Oversampling –Under=sampling Sampling Theorem Aliasing Ideal Reconstruction Folding
3
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
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
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
6 Spectrum for x[n]
7
7 Digital Frequency and Frequency Spectrum
8
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
9 Spectrum (Digital) with Over-sampling
10
10 Spectrum (Digital) with fs = f (under-sampling)
11
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
12 Sampling Theorem
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.