ECTE301_notes_week3 - Digital Signal Processing Week 3...

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Digital Signal Processing Week 3 Sampling: Analog to Digital Conversion
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Motivation Many signals are time-continuous in nature; DSP: processing of digital signals How to convert analog signals to digital ones?
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Motivation The sampling process should not yield any loss of the information. In other words, the original analog signal should be reconstructed (restored) based on the time-discrete sequence. sampling Analog signal Discrete-time sequence ) ( t x ) ( ] [ s nT x n x =
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Motivation The problem is how to choose the sampling interval T s so that the original analog signal can be reconstructed. sampling Analog signal Discrete-time sequence ) ( t x ) ( ] [ s nT x n x =
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Outlines Sampling of a sinusoid; Sampling: spectral perspective Discrete-to-continuous conversion The sampling theorem
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Sampling The sampler takes a snapshot of the x(t) for every T s Analog signal Discrete-time sequence ) ( t x ) ( ] [ s nT x n x =
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Sampling of a sinusoid Given an analog sinusoid ) cos( ) ( φ ϖ+ = t A t x The discrete sequence after sampling is ) ˆ cos( ) cos( ) ( ] [ ϖ + = + = = n A nT A nT x n x s s s s f f T π 2 ˆ = = is called the normalized radian frequency ˆ
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Sampling of a sinusoid What is the difference between x ( t ) and x [ n ]? Radian frequency has the unit of rad/sec; Normalized frequency has the unit of rad— dimensionless quantity; X(t) is a continuous time function. x[n] is just a number sequence carrying no information about the sampling period. ) cos( ) ( φ ϖ+ = t A t x ) ˆ cos( ) cos( ) ( ] [ ϖ + = + = = n A nT A nT x n x s s
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Sampling of a sinusoid
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Sampling of a sinusoid With different sampling frequency, sampling of an analog signal will will different discrete sequence Sampling of different analog signals may yield the same discrete sequence Sampling frequency must be employed in order to reconstruct the original analog signal
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Shannon Sampling Theorem A continuous-time signal x ( t ) with frequencies no higher than f max can be reconstructed exactly from its samples x[ n ]=x( nT s ), if the samples are taken at a rate f s =1/ T s that is greater than 2 f max , which is called Nyquist sampling rate (or frequency)
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Shannon Sampling Theorem Claude Shannon – father of information theory Harry Nyquist
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ECTE301_notes_week3 - Digital Signal Processing Week 3...

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