1. 1 Today's lecture −Concept of Aliasing −Spectrum for Discrete Time Domain −Over-Sampling and Under-Sampling −Aliasing −Folding −Ideal Reconstruction −D-to-A Reconstruction −Pulse Shapes for Reconstruction −Sampling Theorem & Band-limited Signals

2. 2 Storing Digital Sound

3. 3 The Concept of Aliasing Two different cosine signals can be drawn through the same samples x 1 [n] = cos(0.4πn) x 2 [n] = cos(2.4πn) x 2 [n] = cos(2πn + 0.4πn) x 2 [n] = cos(0.4πn) x 2 [n] = x 1 [n]

4. 4 Reconstruction ? Which one ? Figure 4-4

5. 5 Exercise 4.2 −Show that 7cos (8.4πn - 0.2π) is an alias of 7cos (0.4πn - 0.2π). Also find two more frequencies that are aliases of 0.4π rad.

6. 6 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 = 0,+1,+2,… l ooo

7. 7 Spectrum of a Discrete-Time Signal y 1 [n] = 2cos(0.4πn)+ cos(0.6πn) y 2 [n] = 2cos(0.4πn)+ cos(2.6πn)

8. 8 Sampling Theorem

9. 9 Aliasing −Aliasing occurs when we do not sample the signal fast enough that is if f s is not greater than 2f max

10. 10 Ideal Reconstruction −The D-to-C converter gives y(t) = y[n] | n = f s t above substitution only holds true when y(t) is a sum of sinusoids Special case y[n] = A cos(2πf o nT s +  ) Then y[t] = A cos(2πf o t +  ) −What if mathematical formula for y(t) is not known, and only a sequence of numbers for y[n] is known?

11. 11 Actual Reconstruction −D-to-A converter or D-to-C converter must fill-in the values between sample times −Interpolation scheme needs to be used −Discrete-time signal has an infinite number of aliases, + 2 πl, 2 πl - l = integer −Which discrete-time frequency to be used? −The D-to-C converter always selects the lowest possible frequency components (principal alias) - π < < π ooo o

12. 12 Digital Frequency and Frequency Spectrum

13. 13 Spectrum (Digital) with Over-sampling

14. 14 Spectrum (Digital) with fs = f (under-sampling)