Signals and Systems

1. 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 2πgives an alias = 0.4 π + 2 πl l = 0,1,2,3,….. • Another alias x3[n] = cos(1.6πn) x3[n] = cos(2πn - 0.4πn) x3[n] = cos(0.4πn) Since cos (2πn - θ) = cos (θ ) • All aliases maybe obtained as , + 2 πl , 2 πl - l = integer o o o

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 o o o

5. Spectrum of a Discrete-Time Signal y1[n] = 2cos(0.4πn)+ cos(0.6πn) y2[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(ωot+ ) • Spectrum consists of two spectral lines at +ωo with complex amplitudes 1/2A e +jφ • The sampled discrete-time signal x[n] = A cos((ωo/fs)n+ ) x[n] = 1/2Ae+jφe+ j(ωo/ fs )n+ 1/2Ae - jφe - j(ωo/ fs )n • Has two spectrum lines at ώ = +ωo /fs, but it also must contain all the aliases at the following discrete-time frequencies ώ = + ωo /fs+ 2πl l=0, +1, +2,… ώ = - ωo /fs+ 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 fmax can be reconstructed from its samples x(k Ts) if samples taken at rate fs > 2 fmax Nyquist rate = 2 fmax Nyquist frequency = fs/ 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