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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

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Storing Digital Sound

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The Concept of Aliasing

Two different cosine signals can be drawn through the same samples

x1[n] = cos(0.4πn)

x2[n] = cos(2.4πn)

x2[n] = cos(2πn + 0.4πn)

x2[n] = cos(0.4πn)

x2[n] = x1[n]

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Reconstruction? Which one?

Figure 4-4

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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.

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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 aliasx3[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 = 0,+1,+2,…

l

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Spectrum of a Discrete-Time Signal

y1[n] = 2cos(0.4πn)+ cos(0.6πn)

y2[n] = 2cos(0.4πn)+ cos(2.6πn)

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Sampling Theorem

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Aliasing−Aliasing occurs when we do not sample the

signal fast enough that is if fs is not greater than 2fmax

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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(2πfonTs + )

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?

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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

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Digital Frequency and Frequency Spectrum

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Spectrum (Digital) with Over-sampling

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