![(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 ( '](https://image.slidesharecdn.com/applicationoffourierseries-121021230657-phpapp02/85/Application-of-fourier-series-11-320.jpg)
![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](https://image.slidesharecdn.com/applicationoffourierseries-121021230657-phpapp02/85/Application-of-fourier-series-12-320.jpg)
![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.](https://image.slidesharecdn.com/applicationoffourierseries-121021230657-phpapp02/85/Application-of-fourier-series-13-320.jpg)
![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](https://image.slidesharecdn.com/applicationoffourierseries-121021230657-phpapp02/85/Application-of-fourier-series-14-320.jpg)
![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](https://image.slidesharecdn.com/applicationoffourierseries-121021230657-phpapp02/85/Application-of-fourier-series-15-320.jpg)
![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.”](https://image.slidesharecdn.com/applicationoffourierseries-121021230657-phpapp02/85/Application-of-fourier-series-16-320.jpg)
Application of fourier series
- 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