User manual | Nyqist Sampling Theory

Nyquist Sampling
Theorem
By: Arnold Evia
Table of Contents
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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
Bandlimited signal with bandwidth B
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
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
Input
theEquations
function into the
Set up
fourier transform eqs.
T0 is the period of the
func.
Solve
Dn for
Solve
forone
Dnperiod
Consider period from –
T0/2 to T0/2
Only one value: at t=0
Integral equates to 1 as
e-jnw0(0) = 1
Original Function
Substitute
Dn intoAnswer
first equation
Understand
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
Sampling in the Fourier Domain
• Consider a bandlimited signal f(t)
multiplied with an impulse response train
(sampled):
o
o
If the period of the impulse train is insufficient
(T0 > 1/(2B)), aliasing occurs
When T0=1/(2B), T0 is considered the nyquist
rate. 1/T0 is the nyquist frequency
Visual
Representation of Property
Time • Recall that multiplication
in the time
domain is convolution in the frequency
domain:
• As can be seen in the fourier spectra, it is
Freq.
only necessary to extract the fourier
Domain
spectra from one period to reconstruct the
signal!
Domain
.
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=
=
Sampling Cases
• T0>1/(2B)
o
o
o
Undersampling
Distance between copies of F(w)
that overlap happens
Aliasing occurs, and the higher
frequencies of the signal are
corrupted
• T0<=1/(2B)
o
o
o
Oversampling
Distance between copies of F(w) is
sufficient enough to prevent
overlap
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
html

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