![Page 1: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/1.jpg)
DISCRETE-TIME SIGNAL PROCESSINGLECTURE 4 (SAMPLING)
Husheng Li, UTK-EECS, Fall 2012
![Page 2: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/2.jpg)
PERIODIC SAMPLING
Sampling: , where T is the sampling period. In practice, it is done by A/D converter. The sampling operation is generally invertible.
![Page 3: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/3.jpg)
TWO STAGE REPRESENTATION
We represent the sampling procedure in two stages:• Multiplication with an
impulse train with output • Conversion from impulse
train to discrete time sequence
Note: this is a mathematical formulation, not a physical circuit implementation
![Page 4: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/4.jpg)
FREQUENCY-DOMAIN REPRESENTATION
The frequency domain of the post-sampling signal is given by
Assume that the signal has a limited band .
If the sampling frequency satisfies , there will be no overlap.
![Page 5: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/5.jpg)
EXACT RECOVERY
An ideal low pass filter can be used to obtain the exact original signal.
![Page 6: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/6.jpg)
ALIASING
If the inequality is not valid, the frequency copies of signal will overlap, which incurs a distortion called aliasing.
See the example of cosine function.
![Page 7: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/7.jpg)
NYQUIST-SHANNON THEOREM
Theorem: For a band limited signal within band , it is uniquely determined by its samples , if .
![Page 8: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/8.jpg)
EXAMPLE OF SINUSOIDAL SIGNAL
![Page 9: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/9.jpg)
RECONSTRUCTION OF A BANDLIMITED SIGNAL
The reconstruction is given by
![Page 10: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/10.jpg)
INTUITIVE EXPLANATION
It can be used for D/C converter:
![Page 11: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/11.jpg)
DISCRETE-TIME PROCESSING
We can use C/D converter to convert a continuous-time signal to a discrete-time one, process it in a discrete-time system, and then convert it back to continuous time domain.
![Page 12: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/12.jpg)
EXAMPLE: LTI AND LPF
We can use a discrete-time low pass filter (LPF) to do the low pass filtering for continuous time signal.
![Page 13: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/13.jpg)
EXAMPLE: LTI AND LPF
The ideal low pass discrete-time filter with discrete-time cutoff frequency w has the effect of an ideal low pass filter with cutoff frequency w/T.
![Page 14: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/14.jpg)
CONTINUOUS-TIME PROCESSING OF DISCRETE-TIME SIGNALS
We can also use continuous-time system to process discrete-time signals.
![Page 15: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/15.jpg)
RESAMPLING: DOWNSAMPLING
The downsampling implies
![Page 16: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/16.jpg)
INTUITION IN THE FREQUENCY DOMAIN
With aliasing
Without aliasing
![Page 17: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/17.jpg)
DECIMATOR
A general system for downsampling by a factor of M is the one shown above, which is called a decimator.
![Page 18: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/18.jpg)
UPSAMPLING
The upsampling is given by , where L is the integer factor.
![Page 19: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/19.jpg)
EXPANDER
The output of expander is given by .
In the frequency, we have
![Page 20: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/20.jpg)
INTERPOLATOR
It can be shown that the above structure realizes the upsampling and interpolates the signals between samples:
![Page 21: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/21.jpg)
SIMPLE AND PRACTICAL INTERPOLATION
The ideal interpolator is impossible to implement. In practice, we can use a linear interpolator:
![Page 22: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/22.jpg)
TIME AND FREQUENCY OF LINEAR INTERPOLATOR
![Page 23: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/23.jpg)
CHANGING SAMPLING RATE BY A NON-INTEGER FACTOR
The change of sampling rate by a non-integer factor can be realized by the cascade of interpolator and decimator.
![Page 24: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/24.jpg)
THE FREQUENCY INTUITION
![Page 25: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/25.jpg)
MULTIRATE SIGNAL PROCESSING
Multirate techniques refer in general to utilizing upsampling, downsampling, compressors and expanders in a variety of ways to improve the efficiency of signal processing systems.
![Page 26: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/26.jpg)
INTERCHANGE OF FILTERING WITH COMPRESSOR / EXPANDER
The operations of linear filtering and downsampling / upsampling can be exchanged if we modify the linear filter.
![Page 27: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/27.jpg)
MULTISTAGE DECIMATION
The two stage implementation is often much more efficient than a single-stage implementation.
The same multistage principles can also be applied to interpolation
![Page 28: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/28.jpg)
DIGITAL PROCESSING OF ANALOG SIGNALS
In practice, continuous time signals are not precisely band limited, ideal filters cannot be realized, ideal C/D and D/C converters can only be approximated by A/D and D/A converters.
![Page 29: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/29.jpg)
PREFILTERING TO AVOID ALIASING
We can use oversampled A/D to simplify the continuous-time antialiasing filter.
![Page 30: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/30.jpg)
FREQUENCY DOMAIN INTUITION
Key point: the noise is aliased; but the signal is not. Then, the noise can be removed using a sharp-cutoff decimation filter.
![Page 31: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/31.jpg)
A/D CONVERSION
![Page 32: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/32.jpg)
SAMPLE-AND-HOLD
The zero-order-hold system has the impulse response given by
![Page 33: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/33.jpg)
QUANTIZATION
This quantizer is suitable for bipolar signals.
Generally, the number of quantization levels should be a power of tow, but the number is usually much larger than 8.
![Page 34: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/34.jpg)
ILLUSTATION
![Page 35: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/35.jpg)
D/A CONVERSION
The ideal D/A is given byIn practice, we need to use the above structure.
![Page 36: Husheng Li, UTK-EECS, Fall 2012. An ideal low pass filter can be used to obtain the exact original signal](https://reader038.vdocuments.us/reader038/viewer/2022110320/56649cba5503460f94981ea8/html5/thumbnails/36.jpg)
OVERSAMPLING
Oversampling can make it possible to implement sharp cutoff antialiasing filtering by incorporating digital filtering and decimation.
Oversampling and subsequent discrete-time filtering and downsampling also permit an increase in the step size of the quantizer, or equivalently, a reduction in the number of bits required in the A/D conversion.