Lec. 2: Sampling

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Lec. 2:

Sampling

Lecturer: Hooman Farkhani

Department of Electrical Engineering Islamic Azad University of Najafabad Feb. 2016.

Email: [email protected]

In The Name of Almighty

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A/D and D/A Conversion

S/H

Analog out Digital

in

D/A DSP

Smoothing filter

A/D Conversion

D/A Conversion

Reconstruction

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Sampling

 Sampling is a common technical process.

- E.g.: A movie consists out of a sequences of photographs (the samples).

 The sampling process determines the value of a signal on a predetermined frame of time moments.

 Sampling Frequency: f_s

Sampling circuit

Sampled version of input signal Analog input signal

Sampling pulses

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Sampling in Time:

 Uniform Sampling with period T:

 Sampling a signal is equivalent to the mixing of the signal with a train of deltas.

 Note that the multiplication is a non-linear operation : (inherent non-linearity of Sampling Operation)

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Sampling in Frequency Domain:

 Laplace transform of an infinite sequence of deltas is given by:

 Hence we have:

 The equation shows that the spectrum of x∗(nT) is the superposition of infinite replicas of the input spectrum. These replicas are centered at multiples of the sampling frequency being shifted along the f axis by:

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Sampling in Frequency Domain:

(a) Bilateral spectrum of a continuous-time signal. (b) Sampled spectrum by using fs/2 > f_B

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Aliasing Band

(a) Bilateral spectrum of a continuous-time signal. (c) Sampled spectrum by using fs/2 < f_B

Aliasing

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Another example for Aliasing:

Reference: Pelgrom Book

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sampling with 20 Ms/s in the same sampled data sequence (dots)

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Sampling Theorem (Nyquist Criterion)

 The sampling theorem states:

 If the signal is sampled less than this, the recovery process will pr oduce frequencies that are entirely different than in the original sig nal. These “masquerading” signals are called aliases.

In order to recover a signal, the sampling rate must be greater than twice the highest frequency in the signal.

Stated as an equation, f_sample > 2f_a(max) where f_sample = sampling frequency

f_a(max) = highest harmonic in the analog signal

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Anti-Aliasing Filter:

 The anti-aliasing filter is a low-pass filter that limits high frequencies in the input signal to only those that meet the requirements of the sa mpling theorem.

f_c Filtered analog freq uency spectrum

f f_sample

Sampling frequency spectrum

The filter’s cutoff frequency, f_c, should be less than ½ f_sample.

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Anti-Aliasing Filter:

 The anti-aliasing filter protects the information content of the signal.

 Use an antialiasing filter in front of every quantizer to reject unw anted interferences outside of the band of the interest (either

low pass or band-pass)!

 Exercise: Do researches on these topics:

 1. Undersampling the signals

 2. Oversampling the signals

 3. Non-uniform Sampling and its applications

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Classes of sampling

 Nyquist-rate sampling (f_s > 2. f_sig.max)

- Nyquist data converters

- In practice always slightly oversampled.

 Oversampling ((f_s >> 2. f_sig.max)

- Oversampled data converters

- Anti-alias filtering is often trivial

- Oversampling also helps reduce “quantization noise” (<ore lat er)

 Undersampling, subsampling (f_s < 2. f_sig.max)

- Exploit aliasing to mix RF/IF signals down to baseband - see e.g. Pekau & Haslett, JSSC 11/2005.

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From Analog Toward Digital

 After Limiting the bandwidth (BW) of the input signal (using a nti-aliasing filter), the first step in converting a signal to digital form is to use a sample-and-hold circuit.

 This circuit samples the input signal at a rate determined by a clock signal and holds the level on a capacitor until the next clock pulse.

 A positive half-wave from 0-10 V is shown in blue.

 The sample-and-hold circuit produces the staircase representati on shown in red.

 After Sampling A/D converts the data (see next lactures)

0 V 10 V

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Summary

 Must obey sampling theorem f_s > 2·f_sig,max, – Usually dictates anti-aliasing filter

 If sampling theorem is met, continuous time signal can be

recovered from discrete time sequence without loss of information .

 Sample and Hold circuits can be implemented separately or jus t implemented as a part of the ADC. (This will be discussed lat er)

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References

 Chapter 3 of Pelgrom’s book.

 Chapter 1 (1.2) of Maloberti’s book