sampling, quantization -...

Infocommunication

Sampling,

Quantization

- Bálint TÓTH, BME TMIT -

Overview

PPT is for demonstration, not for learning!

Analog signals – problem: noise, distortion

Digital signals – what are the benefits?

We’re going to talk about A/D and D/A conversions

Sampling

– The sampling theorem

– Band limited signals

Quantization

– Quantization noise

– Signal-to-noise ratio (SNR)

Bálint TÓTH – BME TMIT 2

Analog and digital signals

Sampling makes us able to use modern

technology: Audio: CD, MP3, cell phone Video: DVD

Pictures: digital camera, printer Other: Computer, etc.

Sampling


Source: www.wikipedia.org

Effects of sampling (example 1)

Picture



Newspaper



Your eyes


Source: eagleeye.co.ke


Your eyes



Music


Analog-to-digital conversion

Relation between the analog and the sampled

signal (Sampling theorem):

𝑋𝑆 𝑓 = 𝑓𝑠 𝑋(𝑓 − 𝑖 ∗ 𝑓𝑠)

𝑖


Jean Baptiste Joseph Fourier


Source: www.wikipedia.org

Who’s theorem?

The sampling theorem is usually known as the Shannon Sampling

Theorem due to Claude E. Shannon’s paper “A mathematical

theory of communciation” (1948). However, he himself said that

“…is common knowledge in the communication art.”

The minimum required sampling rate fs (i.e. 2xB) is known as the

Nyquist sampling rate or Nyquist frequency because of H.

Nyquist’s work on telegraph transmission in 1924 with K.

Küpfmüller.

The first formulation of the sampling theorem precisely and applied

it to communication is probably a Russian scientist by the name of

V. A. Kotelnikov in 1933.

However, mathematician already knew about this in a different form

and called this the interpolation formula. E. T. Whittaker published

the paper “On the functions which are represented by the

expansions of the interpolation theory” back in 1915.


Digital-to-analog conversion

𝑥 𝑡 → 𝑥𝑘 → 𝑥𝑠 𝑡

The spectrum of a sampled signal:

𝑋𝑆 𝑓 = 𝑥𝑘𝑒−𝑖2𝜋𝑓𝑘𝑇

𝑘

time between samples: T

sampling frequency: fs= 1/T


Band limited signal, low-pass filter

There is no spectral overlapping if

𝑋 𝑓 − 𝑖 ∗ 𝑓𝑆 = 0, 𝑖𝑓 𝑓 < −𝐵 𝑜𝑟 𝑓 > 𝐵, 𝑖 > 0

With ideal low-pass filter the reconstruction is

granted if

𝑓𝑠≥ 2𝐵

(Nyquist frequency)

Bálint TÓTH – BME TMIT

13

Sub- and oversampling


Below the Nyquist-frequency the sampling

points are farther from each other, the signal

cannot be perfectly reconstructed (with low-

pass filter)

– Subsampling

Above the Nyquist-frequency the sampling

points are nearer to each other, the signal can

be better recognised from the samples.

– Oversampling

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Aliasing, leakage

This phenomena is caused by the poor choise

of sampling frequency or by non-ideal filters.

Aliasing: input filter (anti-aliasing filter).

Leakage: output filter.


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Aliasing example


Aliasing example – with antialising


Aliasing example – w/o antialising


Quantization

Convert the signal to discrete values


Source: www.cnx.org

Quantization interval and steps


∆ = 2m / N

20

m

Noise (linear quantization)

Power of noise:

Signal-to-noise ration for sinusoid signal:

SNR = 1.74 + 6.02n [dB]


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Signal transmission and reconstruction with D/A and A/D


Filter Sampler &

holder Quantization Coder

Decoder Filter

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Filter Sampler &


Decoder Filter

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Filter Sampler &


Decoder Filter

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3, 5, 7, 6, 4, 4, 5, 3, 2

Filter Sampler &


Decoder Filter

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3, 5, 7, 6, 4, 4, 5, 3, 2

Filter Sampler &


Decoder Filter

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Filter Sampler &


Decoder Filter

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