Lecture 9: Structure for Discrete-Time System XILIANG LUO 2014/11 1.
Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled...
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Transcript of Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled...
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