16 Sampling The sampling theorem, which is a relatively straightforward consequence of the modulation theorem, is elegant in its simplicity. It basically states that a bandlimited time function can be exactly reconstructed from equally spaced samples provided that the sampling rate is sufficiently high-specifically, that it is greater than twice the highest frequency present in the signal. A similar result holds for both continuous time and discrete time. One of the important consequences of the sampling theorem is that it provides a mechanism for ex- actly representing a bandlimited continuous-time signal by a sequence of samples, that is, by a discrete-time signal. The reconstruction procedure con- sists of processing the impulse train of samples by an ideal lowpass filter. Central to the sampling theorem is the assumption that the sampling fre- quency is greater than twice the highest frequency in the signal. The recon- structing lowpass filter will always generate a reconstruction consistent with this constraint, even if the constraint was purposely or inadvertently violated in the sampling process. Said another way, the reconstruction process will al- ways generate a signal that is bandlimited to less than half the sampling fre- quency and that matches the given set of samples. If the original signal met these constraints, the reconstructed signal will be identical to the original sig- nal. On the other hand, if the conditions of the sampling theorem are violated, then frequencies in the original signal above half the sampling frequency be- come reflected down to frequencies less than half the sampling frequency. This distortion is commonly referred to as aliasing, a name suggestive of the fact that higher frequencies (above half the sampling frequency) take on the alias of lower frequencies. The concept of aliasing is perhaps best understood in the context of sim- ple sinusoidal signals. Given samples of a sinusoidal signal, many continuous- time sinusoids can be threaded through the samples. For example, if the sam- ples were all of equal height, they could correspond to samples of a sinusoid of zero frequency or in fact a sinusoid at any frequency that is an integer mul- tiple of the sampling frequency. From the samples alone there is clearly no way of determining which of the continuous sinusoids was sampled. The re- construction filter, however, makes the assumption that the samples also cor- respond to a frequency less than half the sampling frequency; so for this par- ticular example, the reconstructed output will be a constant. If, in fact, the 16-1
Transcript
16Sampling
The sampling theorem, which is a relatively straightforward consequence ofthe modulation theorem, is elegant in its simplicity. It basically states that abandlimited time function can be exactly reconstructed from equally spacedsamples provided that the sampling rate is sufficiently high-specifically, thatit is greater than twice the highest frequency present in the signal. A similarresult holds for both continuous time and discrete time. One of the importantconsequences of the sampling theorem is that it provides a mechanism for ex-actly representing a bandlimited continuous-time signal by a sequence ofsamples, that is, by a discrete-time signal. The reconstruction procedure con-sists of processing the impulse train of samples by an ideal lowpass filter.
Central to the sampling theorem is the assumption that the sampling fre-quency is greater than twice the highest frequency in the signal. The recon-structing lowpass filter will always generate a reconstruction consistent withthis constraint, even if the constraint was purposely or inadvertently violatedin the sampling process. Said another way, the reconstruction process will al-ways generate a signal that is bandlimited to less than half the sampling fre-quency and that matches the given set of samples. If the original signal metthese constraints, the reconstructed signal will be identical to the original sig-nal. On the other hand, if the conditions of the sampling theorem are violated,then frequencies in the original signal above half the sampling frequency be-come reflected down to frequencies less than half the sampling frequency.This distortion is commonly referred to as aliasing, a name suggestive of thefact that higher frequencies (above half the sampling frequency) take on thealias of lower frequencies.
The concept of aliasing is perhaps best understood in the context of sim-ple sinusoidal signals. Given samples of a sinusoidal signal, many continuous-time sinusoids can be threaded through the samples. For example, if the sam-ples were all of equal height, they could correspond to samples of a sinusoidof zero frequency or in fact a sinusoid at any frequency that is an integer mul-tiple of the sampling frequency. From the samples alone there is clearly noway of determining which of the continuous sinusoids was sampled. The re-construction filter, however, makes the assumption that the samples also cor-respond to a frequency less than half the sampling frequency; so for this par-ticular example, the reconstructed output will be a constant. If, in fact, the
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original signal was a sinusoid at the sampling frequency, then through thesampling and reconstruction process we would say that a sinusoid at a fre-quency equal to the sampling frequency is aliased down to zero frequency(DC).
Thus, as we demonstrate in this lecture, if we sample the output of a sinu-soidal oscillator and then reconstruct with a lowpass filter, as the oscillatorfrequency increases from zero, the output of the lowpass filter will corre-spondingly increase. The output frequency will match the input frequency un-til the oscillator frequency reaches half the sampling frequency. As the oscilla-tor frequency continues to increase, the output of the lowpass filter will beginto decrease in frequency.
It is important to understand that in sampling and reconstruction with anideal lowpass filter, the reconstructed output will not be equal to the originalinput in the presence of aliasing, but samples of the reconstructed output willalways match the samples of the original signal. This relationship is empha-sized in this lecture through a computer movie. It is also important to recog-nize that aliasing is not necessarily undesirable. As we illustrate with a hope-fully enjoyable and entertaining visit with Dr. Harold Edgerton at MIT'sStrobe Laboratory, stroboscopy heavily exploits the concept of aliasing. Spe-cifically, by using pulses of light, motion too fast for the eye to follow can bealiased down to much lower frequencies. In this case, the strobe light repre-sents the sampler, and the lowpass filtering is accomplished visually.
TRANSPARENCY16.5Time waveforms andspectra for a sampledsinusoidal signal, withno aliasing.
p(t) = 6(t-nT)n=- -00
x(t)
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TRANSPARENCY16.6The same asTransparency 16.5,with the frequencyincreasing.
TRANSPARENCY16.7The input frequencyincreased to the pointwhere aliasing occurs.
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TRANSPARENCY16.8An input frequencyconsiderably abovethat at which aliasingoccurs.
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DEMONSTRATION16.1Audio demonstrationof aliasing with asinusoidal oscillator.
TRANSPARENCY16.9Discrete-timeprocessing of DISCRETE-TIME PROCESSINGcontinuous-time OF CONTINUOUS-TIME SIGNALSsignals.
xConversion x[n] Discrete-time y[n] Conversion tods--c--- r-t---im s coninuus-im
TRANSPARENCY16.10Conversion of acontinuous-time signalto a discrete-timesignal interpreted intwo steps. Thecontinuous-time signalis first sampled with aperiodic impulse train,and the impulse trainvalues are thenconverted to adiscrete-timesequence.
C/D conversion
p(t)
Conversion of
x (t) Xx (t) t sr ex[n] =xe(nT)
sequence
DEMONSTRATION16.2Use of a strobe to viewthe motion of anoscillating spring.
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DEMONSTRATION16.3Use of a strobe to viewthe motion of arotating fan.
DEMONSTRATION16.4A rotating diskobserved with astrobe.
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DEMONSTRATION16.5Water drops as seenwith a strobe.
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