• Presented by.,

• S.Shanmathee

Sampling Theorem

2/6/2015

ADC

• Generally signals are analog in nature (eg:speech,weather

signals).

• To process the analog signal by digital means, it is essential

to convert them to discrete-time signal , and then convert

them to a sequence of numbers.

• The process of converting an analog to digital signal is

‘Analog-to-Digital Conversion’.

• The ADC involves three steps which are:

1)Sampling

2)Quantization

3)coding

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• Analog signals: continuous in time and amplitude

– Example: voltage, current, temperature,…

• Digital signals: discrete both in time and amplitude

– Example: attendance of this class, digitizes analog

signals,…

• Discrete-time signal: discrete in time, continuous in

amplitude

– Example: hourly change of temperature in Austin

TYPES OF SIGNALS

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• During sampling process, a continuous-time signal is

converted into discrete -time signals by taking samples

of continuous-time signal at discrete time intervals.

)()( txnTsx

T=Sampling Interval

x (t)=Analog input signal

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•Sampling theorem gives the criteria for minimum number

of samples that should be taken.

•Sampling criteria:-”Sampling frequency must be

twice of the highest frequency”

fs=2W

fs=sampling frequency

w=higher frequency content

2w also known as Nyquist rate 2/6/2015

•Nyquist rate is defined as the minimum sampling rate for the

perfect reconstruction of the continuous time signals from

samples.

•Nyquist rate=2*highest frequency component

=2*W

•So sampling rate must be greater than or equal to nyquist rate

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•There are two parts,

representation of x(t) in its samples

reconstruction of x(t)

Representation of x(t) in its samples

1.Define x∂(t)

2.Take fourier transform of x∂(t)) (i.e) x∂(f)

3.Relation between x(f) and x∂(f)

4.Relation between x(t) and x(nTs)

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Reconstruction of x(t)

1.Take inverse fourier transform of x∂(f)

2.Show that x(t) is obtained back with the help of

interpolation function

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•While providing sampling theorem we considered fs=2W

•Consider the case that fs < 2W

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Effects of Aliasing,

1.Distortion.

2.The data is lost and it cannot be recovered.

To avoid Aliasing,

1.sampling rate must be fs>=2W.

2.strictly bandlimit the signal to ’W’.

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In general form, any continuous signal can be written as

S(t)=A1 cos(jw1t)+ A2 cos(jw2t)+ A3 cos(jw3t)

F1= w1/2∏ = 50∏/2∏ = 25HZ

F2= w2/2∏ = 300∏/2∏ = 150HZ

F3= w3/2∏ = 100∏/2∏ = 50HZ

Here, highest frequency component=150HZ

Hence Nyquist rate=2*150HZ=300HZ

)100cos(10)300sin(20)50cos(5)( tttts

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•What is the minimum sampling rate(nyquist rate)?

Highest frequency=100HZ

So, Nyquist rate=2W=2*100=200HZ•If sampling frequency is 400HZ then what is the discrete

time signal obtained?

f=freq of continuous signal/sampling freq

=100/400=1/4

Discrete time signal=5 cos(2∏fn)=5 cos (2∏*1/4 n)

=5 cos(∏n/2)

)200cos(5)( tts

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“SIGNALS AND SYSTEMS”

by Dr.J.S.Chitode

Optimist: "The glass is

half full."

Pessimist: "The glass is

half empty."

Engineer: "That glass is

twice as large as it

needs to be." 2/6/2015