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Application of fourier series

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Application of fourier series

1. 1. Applicat ion of fourier series inSAMPLINGPresented by: GIRISH DHARESHWAR
2. 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. 3. • Sampling theorem says there is enough information to reconstruct the signal, which means sampled signal looks like original one
4. 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. 5. Where ???IN COMMUNICATIONA AO NL G D ITA IG L D ITA IG L SML G A P IN DP SS NL IG A S NL IG A S NL IG A• Convert analog signals into the digital information-sampling & involves analog-to-digital conversionD 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 TIOS 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. 6. AA GN 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 AS 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. 7. Sampling rate :8 5*sin (2 4t)64 Amplitude = 52 Frequency = 4 Hz0-2-4-6 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. 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. 9. Here sampling rate is 8.5 Hzand 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. 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. 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. 12. The Fourier Transform of (t) is 1. ( t ) exp( i t ) dt exp( i [0]) 1 (t) tAnd the Fourier Transform of 1 is ( ): 1 exp( i t ) dt 2 ( ( ) t
13. 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 tThe function exp(i 0t) is the essential component of Fourier analysis. It isa pure frequency.
14. 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. 15. The Modulation Theorem: The FourierTransform 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. 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 ) dtRearranging: 2 f( ) 1 2 F ( t ) e x p ( i[ ] t) dt 2Relabeling the integration variable from t to ’, we can see that we have aninverse 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. 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