Lecture 4: Sampling [2]XILIANG LUO

2014/10

Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete-time signal as:

Ideal C/D converter

Ideal Sampling Impulse train modulator

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

What about DTFT

This is the general relationship between the periodically sampled sequenceand the underlying continuous time signal

Nyquist-Shannon Sampling Let be a band-limited signal with

Then is uniquely determined by its samples

if:

The frequency is referred to as the Nyquiest frequency

The frequency 2 is called Nyquist rate

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

Band-limited Signal

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:

Process Discrete-Time Signal

Process Discrete-Time Signal

Example: Non-Integer Delay

HW Due on 10/104.31

4.34

4.53

4.60

4.61

4.21

4.54

need multi-rate signal processing knowledge

Next 1. Change sampling rate

2. Multi-rate signal processing

3. Quantization

4. Noise shaping

Change Sampling Rate

Conceptually, we can do this by reconstruct the continuous timesignal first, then resample the reconstructed continuous signal

Sampling Rate Reduction

Down-sampling

Downsampling

Downsampling

Anti-Aliasing Filter

Aliasing Example

Upsampling

Upsampling

Frequency Domain

Upsampling

Filtering Compressor

Filtering Expander

Polyphase DecompositionGoal: efficient implementation structure

k=0,1,…,M-1

Polyphase Decomposition

Polyphase in Freq Domain

Polyphase component filters

Polyphase Filters

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

Polyphase + Decimation Filter

Polyphase + Decimation Filter

Polyphase + Decimation Filter

Polyphase + Interp Filter

Polyphase + Interp Filter

Polyphase + Interp Filter

Ideal

Practical

Avoid Aliasing

Simple Anti-Aliasing Filter

Oversampling C/D

Oversampling C/D

Oversampling C/D Advantages nominal analog filter exact linear phase

A/D Conversion

Zero-order Hold System

Quantization

a Typical Quantizer

Quantization Error

Quantization ErrorAssumptions:

Quantization Error

D/A Conversion

Ideal reconstruction:

D/A Conversion

D/A Conversion

Effect of Quantization:

D/A Conversion

D/A Conversion

compensated filter

D/A Conversion

D/A Conversion

D/A Conversion

Practical D/A Conversion

Practical Digital System