This document contains information about Himanshu Singh, a 7th semester student in the Electronics & Communication department studying Digital Signal Processing. It discusses key topics in digital signal processing including the uniform sampling theorem, Nyquist rate, aliasing, and practical considerations for sampling analog signals. It also provides examples of sampling different frequency signals and the effects of undersampling.

1. Name – Himanshu Singh
Dept – Electronics & Communication
Enroll no. – 131080111003
Sub. – Digital Signal Processing
Sem – 7th

2.  Uniform sampling theorem
A bandlimited signal having no spectral
components above fm hertz can be determined
uniquely by values sampled at uniform intervals
of
sec
2
1
m
S
f
T ≤

3.  A theoretically sufficient condition to allow an
analog signal to be reconstructed completely
from a set of uniformly spaced discrete-time
samples
 Nyquist rate
mS ff 2≥
mS ff 2=
e.g.) speech 8kHz
audio 44.1kHz
fm fs
Speech 3.2kHz
( 사람의 한
계 )
8 kHz
( 휴대폰 )
Audio 20kHz
( 가청주파
수 )
44.1kHz
(mp3)

4. ∑
∞
−∞=
−==
n
SSS nTtnTxtxtxtx )()()()()( δδ
 In the time domain,
 In the frequency domain,∑
∞
−∞=
−=∗=
n
S
S
S nffX
T
fXfXfX )(
1
)()()( δ
∑
∞
−∞=
−=
n
SnTttx )()( δδ
1
( ) ( )S
ns
X f f nf
T
δ δ
∞
=−∞
= −∑

5. FT
FT
FT

6.  The analog waveform can theoretically be
completely recovered from the samples by the use
of filtering (see next figure).
 Aliasing
If , some information will be lost.
 Cf) Practical consideration
 Perfectly bandlimited signals do not occur in nature.
 A bandwidth can be determined beyond which the
spectral components are attenuated to a level that is
considered negligible.
mS ff 2<

7. 2sf W<
2sf W>
2sf W<

8.  Due to undersampling
 Appear in the frequency band between
mms fff and)( −

9.  Helicopter 100Hz
 Sampling at 220Hz  100Hz
 Sampling at 80Hz  20Hz
 How about sampling at 120Hz?
f0 100
-100
f0 100
-100
320
120
-120
-320
540
340
-340
f
100
-100
20
-20-180
-60
60
-140

10.  Effect in the time domain

11.  More practical method
 A replication of X(f), periodically repeated in
frequency every fs Hz.
 Weighted by the Fourier series coefficients of the
pulse train, compared with a constant value in the
impulse-sampled case.
∑
∞
−∞=
==
n
tnfj
npS
S
ectxtxtxtx π2
)()()()(
∑∑
∞
−∞=
∞
−∞=
−==
n
Sn
n
tnfj
nS nffXcectxFfX S
)(})({)( 2π
)sinc(
1
ss
n
T
nT
T
c =


 +<≤−
=
otherwise,0
2/2/,/1
)(
TnTtTnTT
tx ss
p

12. 1/T-1/T
1/T-1/T

13.  The simplest and most popular sampling method
 Significant attenuation of the higher-frequency
spectral replicates
 The effect of the nonuniform spectral gain P(f)
applied to the desired baseband spectrum can be
reduced by postfiltering operation.
∑
∞
−∞=
−∗=∗=
n
SSS nTtnTxtptxtxtptx )()()()()()()( δδ
∑
∞
−∞=
−=∗=
n
S
S
S nffX
T
fPfXfXfPfX )(
1
)()()()()( δ

14.  the most economic solution
 Analog processing is much more costly than
digital one.