IT SHOWS THE SAMPLING OF DISCRETE TIME SIGNAL AND ITS RECONSTRUCTION

1. Sampling and
reconstruction of
discrete time signal
BY :- KARAN SATI

2. Discrete time signal
• A discrete signal or discrete-time signal is a time series consisting of
a sequence of quantities.
• Discrete-time signals may have several origins, but can usually be
classified into one of two groups:-
• By acquiring values of an analog signal at constant or variable rate.
This process is called samling
• By observing an inherently discrete-time process, such as the weekly
peak value of a particular economic indicator.

3. Sampling :-
• Sampling is a process of breakage of continuous signal to
discrete signal.
• The output of system is recorded at different intervals of
time.
• Time interval between two consecutive sampling intervals
is called Sampling period or Sample interval.
• The sampling frequency or sampling rate, fs, is the average
number of samples obtained in one second (samples per
second), thus fs = 1/T.

4. Sampling theorem
 The sampling theorem was published in 1949 in a scientific article of C.E. Shannon
Let x(t) be a band (frequency)-limited signal
X(jw) = 0 for |w|>wM.
Then x(t) is uniquely determined by its samples {x(nT)} when the sampling frequency satisfies:
Ms ww 2
2wM is known as the Nyquist rate, as it represents the largest frequency
that can be reproduced with the sample time.

5. Sampling Effects: Frequency Domain
Xc(j)

N-N
XS(j)

N-N S-S 2S-2S

S-S 2S-2S
S > 2 N
S < 2 N (aliasing)
FourierTransform of
continuous function
FourierTransform of sampled
function

6. Reconstruction

7. Reconstruction :-
• In signal processing, reconstruction usually means the
determination of an original continuous signal from a sequence of
equally spaced samples.

8. Reconstruction
• Since spectrum of sampled signal consists of baseband spectrum
and spectral images shifted at multiples of 2π/T, reconstruction
means isolating the baseband image
• Concept: lowpass filter to pass baseband while removing images
XS(j)

N-N S-S 2S-2S

9. Reconstruction :
• Multiplication by rectangular pulse in frequency
domain (LPF) corresponds to convolution by sinc( )
function in time domain
• Because sinc( ) is non-causal and of infinite extent,
practical reconstruction requires an approximation
to the ideal case

10. Reconstruction
The process of reconstructing a continuous time
signal x(t) from its samples is known as interpolation.
Reconstruction is accomplished by passing the sampled signal
through an ideal low pass filter.

11. 1
1/T
T
1
When the sampling frequency ws is less
than twice the band-limited frequency wM,
there is no overlaps the spectrum X(jw)
If this is true, the original signal x(t) can be
recovered from the impulse sampled xp(t),
by passing it through a low pass filter H(jw)
with gain T and cutoff frequency
between wM and ws-wM.

12. Thank you