Todays lecture Concept of Aliasing Spectrum for Discrete
![Spectrum of a Discrete-Time Signal y 1[n] = 2 cos(0. 4πn)+ cos(0. 6πn) y](https://slidetodoc.com/presentation_image/50615a4153f4bd92262ac6653b3779dc/image-7.jpg "Spectrum of a Discrete-Time Signal y 1[n] = 2 cos(0. 4πn)+ cos(0. 6πn) y")
![Ideal Reconstruction − The D-to-C converter gives y(t) = y[n] |n = fs t](https://slidetodoc.com/presentation_image/50615a4153f4bd92262ac6653b3779dc/image-10.jpg "Ideal Reconstruction − The D-to-C converter gives y(t) = y[n] |n = fs t")
- Slides: 14
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
Storing Digital Sound
The Concept of Aliasing Two different cosine signals can be drawn through the samples x 1[n] = cos(0. 4πn) x 2[n] = cos(2πn + 0. 4πn) x 2[n] = cos(0. 4πn) x 2[n] = x 1[n]
Reconstruction? Which one? Figure 4 -4
Exercise 4. 2 − Show that 7 cos (8. 4πn - 0. 2π) is an alias of 7 cos (0. 4πn - 0. 2π). Also find two more frequencies that are aliases of 0. 4π rad.
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 lalias 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, … o o o
Spectrum of a Discrete-Time Signal y 1[n] = 2 cos(0. 4πn)+ cos(0. 6πn) y 2[n] = 2 cos(0. 4πn)+ cos(2. 6πn)
Sampling Theorem
Aliasing − Aliasing occurs when we do not sample the signal fast enough that is if fs is not greater than 2 fmax
Ideal Reconstruction − The D-to-C converter gives y(t) = y[n] |n = fs t above substitution only holds true when y(t) is a sum of sinusoids Special case y[n] = A cos(2πfon. Ts + ) Then y[t] = A cos(2πfot + ) − What if mathematical formula for y(t) is not known, and only a sequence of numbers for y[n] is known?
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 , o + 2 πl , 2 πl l = integer o o − Which discrete-time frequency to be used? − The D-to-C converter always selects the lowest possible frequency components (principal alias) -π < <π o
Digital Frequency and Frequency Spectrum
Spectrum (Digital) with Over-sampling
Spectrum (Digital) with fs = f (under-sampling)
