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Lecture 4: Sampling [2] XILIANG LUO 2014/10

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Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as: Ideal C/D converter

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Ideal Sampling Impulse train modulator

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Fourier Transform of Ideal Sampling Fourier Transform of periodic impulse train is an impulse train:

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What about DTFT This is the general relationship between the periodically sampled sequence and the underlying continuous time signal

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Nyquist-Shannon Sampling

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Process Cont. Signal A main application of discrete-time systems is to process continuous- time signal in discrete-time domain

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Band-limited Signal

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Observations For band-limited signal, we are processing continuous time signal using discrete-time signal processing For band-limited signal, the overall system behaves like a linear time- invariant continuous-time system with the following frequency domain relationship:

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Process Discrete-Time Signal

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Example: Non-Integer Delay

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HW Due on 10/10 4.31 4.34 4.53 4.60 4.61 4.21 4.54 need multi-rate signal processing knowledge

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Next 1. Change sampling rate 2. Multi-rate signal processing 3. Quantization 4. Noise shaping

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Change Sampling Rate Conceptually, we can do this by reconstruct the continuous time signal first, then resample the reconstructed continuous signal

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Sampling Rate Reduction Down-sampling

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Downsampling

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Anti-Aliasing Filter

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Aliasing Example

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Upsampling

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Frequency Domain

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Upsampling

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Filtering Compressor

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Filtering Expander

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Polyphase Decomposition Goal: efficient implementation structure k=0,1,…,M-1

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Polyphase Decomposition

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Polyphase in Freq Domain Polyphase component filters

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Polyphase Filters y[n]=x[n]*h[n]

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Polyphase + Decimation Filter

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Polyphase + Interp Filter

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Ideal

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Practical

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Avoid Aliasing

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Simple Anti-Aliasing Filter

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Oversampling C/D

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Advantages nominal analog filter exact linear phase

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A/D Conversion

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Zero-order Hold System

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Quantization

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a Typical Quantizer

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Quantization Error

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Assumptions:

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Quantization Error

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D/A Conversion Ideal reconstruction:

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D/A Conversion

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Effect of Quantization:

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D/A Conversion

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compensated filter

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D/A Conversion

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Practical D/A Conversion

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Practical Digital System

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