# Nyquist Sampling Theorem

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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))

Bandlimited signal with bandwidth B

• 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

ISnpeuttuthpeEfuqnucatiotinoninsto the

fourier transform eqs.

T0 is the period of the

func.

SolSveoDlvnefofor ronDenperiod

Consider period from – T0/2 to T0/2 Only one value: at t=0

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

SubUstnitduteersDtnainndtoAfirnsst weqeuration

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

Original Function

Fourier Spectra

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

. = • Visual

Time

RReperecsaelnltatthioantomf Purolpteiprtylication

in

the

time

Domain 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!

=

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

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))

Bandlimited signal with bandwidth B

• 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

ISnpeuttuthpeEfuqnucatiotinoninsto the

fourier transform eqs.

T0 is the period of the

func.

SolSveoDlvnefofor ronDenperiod

Consider period from – T0/2 to T0/2 Only one value: at t=0

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

SubUstnitduteersDtnainndtoAfirnsst weqeuration

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

Original Function

Fourier Spectra

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

. = • Visual

Time

RReperecsaelnltatthioantomf Purolpteiprtylication

in

the

time

Domain 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!

=

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

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