1. Applicat
ion of
fourier
series
in
SAMPLING
Presented by:
GIRISH DHARESHWAR

2. WHAT IS SAMPLING ?
• It is the process of taking the
samples of the signal at intervals
Aliasing
cannot distinguish between
higher and lower frequencies
Sampling theorem:
 to avoid aliasing, sampling rate
must be at least twice the
maximum frequency component
(`bandwidth’) of the signal

3. • Sampling theorem says
there is enough
information to reconstruct
the signal, which means
sampled signal looks like
original one

4. Why ??????????
• Most signals are analog in
nature, and have to be sampled
 loss of information
• Eg :Touch-Tone system of
telephone dialling, when button
is pushed two sinusoid signals
are generated (tones) and
transmitted, a digital system speech signal
determines the frequences and
uniquely identifies the button –
digital

5. Where ???IN COMMUNICATION
A AO
NL G D ITA
IG L D ITA
IG L
SML G
A P IN DP
S
S NL
IG A S NL
IG A S NL
IG A
• Convert analog signals into the digital information-
sampling & involves analog-to-digital conversion
D ITA
IG L D ITA
IG L A AO
NL G
DP
S S NL
IG A
R C N TR C N
E O S U TIO
S NL
IG A S NL
IG A
convert the digital information, after being processed
back to an analog signal
• involves digital-to-analog conversion & reconstruction
e.g. text-to-speech signal (characters are used to
generate artificial sound)

6. AA G
N LO D ITA
IG L AA G
N LO
D ITA
IG L
S MP G
A LIN S NL
IG A
DP
S S NL
IG A
R C N TR C N
E O S U TIO
S NL
IG A
S NL
IG A
 perform both A/D and D/A conversions
 e.g. digital recording and playback of music (signal is
sensed by microphones, amplified, converted to digital,
processed, and converted back to analog to be played

7. Sampling rate :
8
5*sin (2 4t)
6
4
Amplitude = 5
2 Frequency = 4 Hz
0
-2
-4
-6
We take an
-8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ideal sine wave
to discuss
effects of
A sine wave
sampling

8. A sine wave signal and correct sampling
8
5*sin(2 4t)
6
Amplitude = 5
4
2
Frequency = 4 Hz
0 Sampling rate = 256
samples/second
-2
Sampling duration =
-4
1 second
-6
We do sampling of 4Hz
-8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
with 256 Hz so sampling
seconds
is much higher rate than
the base frequency, good
Thus after sampling we can reconstruct
the original signal

9. Here sampling rate is 8.5 Hz
and the frequency is 8 Hz
An undersampled signal
Sampling rate
Red dots
2
sin(2 8t), SR = 8.5 Hz represent the
sampled data
1.5
1
0.5
0
Undersampling
-0.5
can be confusing
-1 Here it suggests
a different
-1.5
frequency of
-2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
sampled signal
 Loss of information

10. The Discrete Time Fourier Transform
(DTFT) and its Inverse :
• The Fourier transform is an equation to
calculate the frequency, amplitude and phase
of each sampled signal needed to make up
any given signal f(t):
F ( ) f (t ) e x p ( i t ) dt
1
f (t ) F ( ) ex p (i t) d
2

11. (t)
function Properties
t
(t ) d t 1
(t a ) f (t ) d t (t a ) f (a ) dt f (a )
ex p ( i t ) d t 2 (
ex p [ i ( ') t ] d t 2 ( '

12. The Fourier Transform of (t) is 1.
( t ) exp( i t ) dt exp( i [0]) 1
(t)
t
And the Fourier Transform of 1 is ( ): 1 exp( i t ) dt 2 (
( )
t

13. The Fourier transform of exp(i 0 t)
F exp( i 0
t) exp( i 0
t ) exp( i t ) dt
exp( i [ 0
] t ) dt 2 ( 0
)
exp(i 0t)
F {exp(i 0t)}
Im t
Re t
The function exp(i 0t) is the essential component of Fourier analysis. It is
a pure frequency.

14. The Fourier transform of cos( t)
F cos( 0
t) cos( 0
t ) exp( i t ) dt
1
exp( i 0
t) exp( i 0
t ) exp( i t ) dt
2
1 1
exp( i [ 0
] t ) dt exp( i [ 0
] t ) dt
2 2
( 0
) ( 0
)
cos( 0t) F {cos( t )}
0
t

15. The Modulation Theorem: The Fourier
Transform of E(t) cos( 0 t)
F E ( t ) cos( 0t ) E ( t ) cos( 0t ) exp( i t ) dt
1
E ( t ) exp( i 0t ) exp( i 0t ) exp( i t ) dt
2
1 1
E ( t ) exp( i [ 0 ] t ) dt E ( t ) exp( i [ 0 ]t) dt
2 2
1  1 
F E ( t ) cos( 0t ) E( 0) E( 0)
2 2
F E ( t ) cos( 0t )
If E(t) = (t), then:
- 0 0

16. The Fourier transform and its inverse are symmetrical:
f(t) -> F( ) and F(t) -> f( ) (almost).
If f(t) Fourier transforms to F( ), then F(t) Fourier transforms to:
F (t ) ex p ( i t ) dt
Rearranging: 2 f( )
1
2 F ( t ) e x p ( i[ ] t) dt
2
Relabeling the integration variable from t to ’, we can see that we have an
inverse Fourier transform:
1
2 F( ) exp( i[ ] )d
2
2 f( )
This is why it is often said that f and F are a “Fourier Transform Pair.”

17. Summary
• Fourier analysis for periodic functions focuses on the
study of Fourier series
• The Fourier Transform (FT) is a way of transforming a
continuous signal into the frequency domain
• The Discrete Time Fourier Transform (DTFT) is a
Fourier Transform of a sampled signal
• The Discrete Fourier Transform (DFT) is a discrete
numerical equivalent using sums instead of integrals
that can be computed on a digital computer
• As one of the applications DFT and then Inverse DFT
(IDFT) can be used to compute standard convolution
product and thus to perform linear filtering