Ece141lab Sampling Lecture

8/11/2019 Ece141lab Sampling Lecture

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ECE 141Sampling Lecture (Ref. Communication Systems by Haykin, 4 th Edition)

1. The Dirac Comb Function in the Frequency DomainDirac Delta Function:

Dirac Comb Function:

Since the Dirac Comb is periodic, we can apply the Fourier Series.

where the Fourier coefficients can be solved from using

Solving for the Fourier coefficients of the Dirac Comb,

Therefore,

Now, taking the Fourier Transform,

What is the Fourier Transform of ?Consider a delta function in the frequency domain, . Taking the inverse Fourier Transform of this yields,

Therefore, the Fourier Transform of is .Now, consider a constant function in the time domain, .We already know that the Fourier Transform of c(t) is but continuing on applying the definition of the FourierTransform to yields

Therefore, it should follow that


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Going back to our initial problem of determining the Fourier Transform of and applying the frequency shiftingproperty of the Fourier Transform,

Substituting this back to the Fourier Transform of the Dirac Comb,

The result is also a Dirac Comb.

The Fourier Transform of a Dirac Comb is also a Dirac Comb.

2. Ideal SamplingIdeal sampling is the process of multiplying the signal (bandwidth W) with a Dirac Comb in the time domain withsampling period , to yield .

is referred to as the ideal sampled signal.In the frequency domain, this is equivalent to

The ideal sampling process is illustrated in the next figure.


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The result of the convolution is

The equation above is called the interpolation formula for reconstructing the original signal from the sequence ofsample values .

4. Sampling TheoremThe sampling theorem for strictly band-limited signals states that,1. A band-limited signal of finite energy, which has no frequency components higher than W Hertz, is completely

described by specifying the values of the signal at instants of time separated by 1/2W seconds.2. A band-limited signal of finite energy, which has no frequency components higher than W Hertz, may be completely

recovered from a knowledge of its samples taken at a rate of 2W samples per second.The minimum sampling rate of 2W is called the Nyquist rate, and 1/2W is called the Nyquist Interval.

In practice, signals of interest are not strictly band-limited. Hence, the sampling process produces aliasing (dark grayareas in the following figure):

To prevent aliasing, the signal is passed into a low-pass anti-aliasing filter to attenuate the high frequencycomponents that are not essential to the information being conveyed by the signal.

Also, practical reconstruction filters does not have an infinitely steep transition from the passband to the stopband. Toaccommodate this transition width, the sampling frequency must be higher than 2W.

5. Non-ideal Sampling5.1. Sample and Hold/Flat-Top Sampling/Pulse Amplitude Modulation (PAM)In practical sampling, the duration of the samples in time are not infinitesimally small but rather, the sampled signalfollows the following waveform:


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The process is called sample and hold (also flat-top sampling and pulse amplitude modulation (PAM)). Flat-top samplingcan be thought of as two steps:1. Instantaneous sampling that which occurs in ideal sampling and,2. Lengthening the duration of each sample.This is illustrated as follows:

Time Domain Frequency Domain

* X

= =

In the time domain, this is equivalent to convolving with , a rectangular pulse with duration :

In the frequency domain,

where takes the form of the sinc function.

Now, to reconstruct the original signal from the sample and hold signal, first we need to pass through a low pass

filter such that we are left with


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We pass this resulting signal through another low pass filter that reverses the effects of . This filter is called anequalizer.We note that

Where the exponential term is due to not being centered at 0 s. This causes the aperture effect amplitudedistortion and a delay of to the original signal.

In order to correct the aperture effect, the response of the equalizer must be equal to the reciprocal of . Taking themagnitude response,

It can be observed from the above that as the value of decreases, the aperture effect also decreases. In fact, for

, amplitude distortion is less than 0.5% and equalization can be omitted altogether.

5.2. Natural SamplingA variation of the sampling process is illustrated in the following figure:

In contrast to sample and hold, the sample is not held for the duration of each pulse but rather, the pulse takes theshape of the signal. This process is called natural sampling. A similar analysis as in that of sample and hold is applicableto this sampling process.

Ece141lab Sampling Lecture

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