Nyquist Sampling Theorem By: Arnold Evia

Table of Contents • What is the Nyquist Sampling Theorem? • Bandwidth • Sampling • Impulse Response Train • Fourier Transform of Impulse Response Train • Sampling in the Fourier Domain

o Sampling cases

• Review

What is the Nyquist Sampling Theorem? • Formal Definition:

o If the frequency spectra of a function x(t) contains no frequencies higher than B hertz, x(t) is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.

• In other words, to be able to accurately reconstruct a signal, samples must be recorded every 1/(2B) seconds, where B is the bandwidth of the signal.

Bandwidth

• There are many definitions to bandwidth depending on the application

• For signal processing, it is referred to as the range of frequencies above 0 (|F(w)| of f(t))

• Signals that have a definite value for the highest frequency are bandlimited (|F(w)|=0 for |w|>B)

• In reality, signals are never bandlimited o In order to be bandlimited, the signal

must have infinite duration in time

Bandlimited signal with bandwidth B

Non-bandlimited signal (representative of real signals)

Sampling

• Sampling is recording values of a function at certain times

• Allows for transformation of a continuous time function to a discrete time function

• This is obtained by multiplication of f(t) by a unit impulse train

Impulse Response Train • Consider an impulse train: • Sometimes referred to as comb function • Periodic with a value of 1 for every nT0, where n is integer

values from -∞ to ∞, and 0 elsewhere

Fourier Transform of Impulse Train

Set up Equations Input the function into the fourier transform eqs. T0 is the period of the func.

Solve for Dn Solve Dn for one period

Consider period from –T0/2 to T0/2

Only one value: at t=0

Integral equates to 1 as e-jnw0(0) = 1

Understand Answer Substitute Dn into first equation

The fourier spectra of the function has an amplitude of 1/T0 at nw0 for values of n from –∞ to +∞, and 0 elsewhere

Distance between each w0 is dependent on T0. Decreasing T0, increases the w0 and distance

Fourier Spectra Original Function

Visual Representation of Property

Sampling in the Fourier Domain

• Consider a bandlimited signal f(t) multiplied with an impulse response train (sampled): o If the period of the impulse train is insufficient

(T0 > 1/(2B)), aliasing occurs o When T0=1/(2B), T0 is considered the nyquist

rate. 1/T0 is the nyquist frequency

• Recall that multiplication in the time domain is convolution in the frequency domain:

• As can be seen in the fourier spectra, it is only necessary to extract the fourier spectra from one period to reconstruct the signal!

. =

= *

Time Domain

Freq. Domain

Sampling Cases • T0>1/(2B)

o Undersampling o Distance between copies of F(w)

that overlap happens o Aliasing occurs, and the higher

frequencies of the signal are corrupted

• T0<=1/(2B) o Oversampling o Distance between copies of F(w) is

sufficient enough to prevent overlap

o Spectra can be filtered to accurately reconstruct signal

Review • Nyquist sampling rate is the rate which samples of the

signal must be recorded in order to accurately reconstruct the sampled signal o Must satisfy T0 <= 1/(2B); where T0 is the time between

recorded samples and B is the bandwidth of the signal

• A signal sampled every T0 seconds can be represented as:

where Ts = T0

Review (cont.) • One way of understanding the importance of the Nyquist

sampling rate is observing the fourier spectra of a sampled signal

• A sampled signal’s fourier spectra is a periodic function of the original unsampled signal’s fourier spectra o Therefore, it is only necessary to extract the data from one

period to accurately reconstruct the signal

• Aliasing can occur if the sampling rate is less than the Nyquist sampling rate o There is overlap in the fourier spectra, and the signal cannot be

accurately reconstructed (Undersampling)

References

Some basic resources can be found here: • http://www.cs.cf.ac.uk/Dave/Multimedia/node149.html • http://www.youtube.com/watch?v=7H4sJdyDztI

ARC website: • http://iit.edu/arc/

ARC BME schedule: • http://iit.edu/arc/tutoring_schedule/biomedical_engineering.s

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