lecture 4: sampling [2] xiliang luo 2014/10. periodic sampling a continuous time signal is sampled...
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![Page 1: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/1.jpg)
Lecture 4: Sampling [2]XILIANG LUO
2014/10
![Page 2: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/2.jpg)
Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete-time signal as:
Ideal C/D converter
![Page 3: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/3.jpg)
Ideal Sampling Impulse train modulator
![Page 4: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/4.jpg)
Fourier Transform of Ideal SamplingFourier Transform of periodic impulse train is an impulse train:
![Page 5: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/5.jpg)
What about DTFT
This is the general relationship between the periodically sampled sequenceand the underlying continuous time signal
![Page 6: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/6.jpg)
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
![Page 7: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/7.jpg)
Process Cont. Signal A main application of discrete-time systems is to process continuous-time signal in discrete-time domain
![Page 8: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/8.jpg)
Band-limited Signal
![Page 9: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/9.jpg)
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:
![Page 10: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/10.jpg)
Process Discrete-Time Signal
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Process Discrete-Time Signal
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Example: Non-Integer Delay
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HW Due on 10/104.31
4.34
4.53
4.60
4.61
4.21
4.54
need multi-rate signal processing knowledge
![Page 14: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/14.jpg)
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 timesignal first, then resample the reconstructed continuous signal
![Page 16: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/16.jpg)
Sampling Rate Reduction
Down-sampling
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Downsampling
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Downsampling
![Page 19: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/19.jpg)
Anti-Aliasing Filter
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Aliasing Example
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Upsampling
![Page 22: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/22.jpg)
Upsampling
![Page 23: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/23.jpg)
Frequency Domain
![Page 24: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/24.jpg)
Upsampling
![Page 25: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/25.jpg)
Filtering Compressor
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Filtering Expander
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Polyphase DecompositionGoal: efficient implementation structure
k=0,1,…,M-1
![Page 28: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/28.jpg)
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 + Decimation Filter
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Polyphase + Decimation Filter
![Page 34: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/34.jpg)
Polyphase + Interp Filter
![Page 35: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/35.jpg)
Polyphase + Interp Filter
![Page 36: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/36.jpg)
Polyphase + Interp Filter
![Page 37: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/37.jpg)
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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Oversampling C/D
![Page 43: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/43.jpg)
Oversampling C/D Advantages nominal analog filter exact linear phase
![Page 44: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/44.jpg)
A/D Conversion
![Page 45: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/45.jpg)
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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Quantization ErrorAssumptions:
![Page 50: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/50.jpg)
Quantization Error
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D/A Conversion
Ideal reconstruction:
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D/A Conversion
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D/A Conversion
Effect of Quantization:
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D/A Conversion
![Page 55: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/55.jpg)
D/A Conversion
compensated filter
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D/A Conversion
![Page 57: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/57.jpg)
D/A Conversion
![Page 58: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/58.jpg)
D/A Conversion
![Page 59: Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling A continuous time signal is sampled periodically to obtain a discrete- time signal as:](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649da05503460f94a8bc1b/html5/thumbnails/59.jpg)
Practical D/A Conversion
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Practical Digital System