The document discusses sampling theory and analog-to-digital conversion. It begins by explaining that most real-world signals are analog but must be converted to digital for processing. There are three steps: sampling, quantization, and coding. Sampling converts a continuous-time signal to a discrete-time signal by taking samples at regular intervals. The sampling theorem states that the sampling frequency must be at least twice the highest frequency of the sampled signal to avoid aliasing. Finally, it provides an example showing how to calculate the minimum sampling rate, or Nyquist rate, given the highest frequency of a signal.

1. • Presented by.,
• S.Shanmathee
Sampling Theorem

2. 2/6/2015

3. 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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4. • 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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5. • 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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6. •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
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7. •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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8. •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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9. 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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10. •While providing sampling theorem we considered fs=2W
•Consider the case that fs < 2W
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11. 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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13. 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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14. •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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15. 2/6/2015
“SIGNALS AND SYSTEMS”
by Dr.J.S.Chitode

16. Optimist: "The glass is
half full."
Pessimist: "The glass is
half empty."
Engineer: "That glass is
twice as large as it
needs to be."
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