Lecture 4:
Sampling [2]
XILIANG LUO
2014/10
Periodic Sampling
 A continuous time signal is sampled periodically to obtain a discretetime signal as:
Ideal C/D converter
Ideal Sampling
 Impulse train modulator
Fourier Transform of
Ideal Sampling
Fourier Transform of periodic impulse train is an impulse train:
What about DTFT
This is the general relationship between the periodically sampled sequence
and the underlying continuous time signal
Nyquist-Shannon Sampling
 Let 𝑥𝑐 (𝑡) be a band-limited signal with
Then 𝑥𝑐 (𝑡) is uniquely determined by its samples
if:
 The frequency Ω𝑁 is referred to as the Nyquiest frequency
 The frequency 2Ω𝑁 is called Nyquist rate
Process Cont. Signal
 A main application of discrete-time systems is to process continuoustime 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 timeinvariant continuous-time system with the following frequency domain
relationship:
Process Discrete-Time Signal
Process Discrete-Time Signal
Example: Non-Integer Delay
HW Due on 10/10
4.21
4.31
4.34
4.53
4.60
4.61
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 time
signal 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 Decomposition
Goal: 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 Error
Assumptions:
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