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### Tele3113 wk1tue

1. 1. TELE3113 Analogue & DigitalCommunications Review of Fourier Transform p. 1
2. 2. Signal Representations(t) = A sin(2π fo t +φo ) or A sin(ωo t +φo ) Time-domain: waveform A: Amplitude Time (seconds) f : Frequency (Hz) (ω=2πf) φ : Phase (radian or degrees) Period (seconds)S(f) Frequency-domain: spectrum fo Frequency (Hz) p. 2
3. 3. Energy and Power of SignalsFor an arbitrary signal f(t), the total energy normalized to unitresistance is defined as ∆ T E = lim ∫ f (t ) 2 dt joules, T →∞ −Tand the average power normalized to unit resistance is defined as ∆ 1 T P = lim T → ∞ 2T ∫ −T f (t ) 2 dt watts ,• Note: if 0 < E < ∞ (finite) P = 0.• When will 0 < P < ∞ happen? p. 3
4. 4. Periodic SignalA signal f(t) is periodic if and only if f (t + T0 ) = f (t ) for all t (*)where the constant T0 is the period.The smallest value of T0 such that equation (*) is satisfied isreferred to as the fundamental period, and is hereafter simplyreferred to as the period.Any signal not satisfying equation (*) is called aperiodic. p. 4
5. 5. Deterministic & Random SignalsDeterministic signal can be modeled as a completely specifiedfunction of time.Example f (t ) = A cos( ω 0 t + θ )Random signal cannot be completely specified as a function oftime and must be modeled probabilistically. p. 5
6. 6. SystemMathematically, a system is a rule used for assigning a function g(t)(the output) to a function f(t) (the input); that is, g(t) = h{ f(t) }where h{•} is the rule or we call the impulse response. f(t) h(t) g(t)For two systems connected in cascade, the output of the first systemforms the input to second, thus forming a new overall system: g(t) = h2 { h1 [ f(t) ] } = h{ f(t) } p. 6
7. 7. Linear SystemIf a system is linear then superposition applies; that is, if g1(t) = h{ f1(t) }, and g2(t) = h{ f2(t) }then h{ a1 f1(t) + a2 f2(t) } = a1 g1(t) + a2 g2(t) (*)where a1, a2 are constants. A system is linear if it satisfiesEq. (*); any system not meeting these requirement is nonlinear. p. 7
8. 8. Time-Invariant and Time-VaryingA system is time-invariant if a time shift in the input resultsin a corresponding time shift in the output so that g (t − t 0 ) = h{ f (t − t 0 )} for any t 0 .The output of a time-invariant system depends on time differences andnot on absolute values of time.Any system not meeting this requirement is said to be time-varying. p. 8
9. 9. Fourier SeriesA periodic function of time s(t) with a fundamental period of T0 can berepresented as an infinite sum of sinusoidal waveforms. Suchsummation, a Fourier series, may be written as: ∞ 2 π nt ∞ 2 πnt s (t ) = A0 + ∑ An cos + ∑ B n sin , (1) n =1 T0 n =1 T0where the average value of s(t), A0 is given by 1 T20 A0 = T0 ∫− T20 s (t ) dt , (2)while 2 T0 2 π nt An = ∫ (3) 2 T0 s (t ) cos dt , T0 − 2 T0and 2 T0 2 π nt Bn = ∫ 2 T0 s (t ) sin dt . (4) T0 − 2 T0 p. 9
10. 10. Fourier SeriesAn alternative form of representing the Fourier series is ∞  2 πnt  s (t ) = C 0 + ∑ C n cos   − φn   (5) n =1  T0 where C0 = A0 , (6) 2 2 Cn = An + B n , (7) B φ n = tan −1 n . (8) AnThe Fourier series of a periodic function is thus seen to consist of asummation of harmonics of a fundamental frequency f0 = 1/T0.The coefficients Cn are called spectral amplitudes, which represent theamplitude of the spectral component Cn cos(2πnf0t − φn) at frequencynf0. p. 10
11. 11. Fourier SeriesThe exponential form of the Fourier series is used extensively incommunication theory. This form is given by ∞ j 2 π nt s (t ) = ∑S n = −∞ n e T0 , (9)where 1 T0 − j 2 π nt (10) Sn = ∫ s (t ) e dt 2 T0 T0 T0 − 2Note that Sn and S−n are complex conjugate of one another, that is S n = S −n . * (11)These are related to the Cn by C n − jφ n (12) S0 = C0 , Sn = e . 2 p. 11
12. 12. Fourier Series Amplitude Spectra (Line Spectra) Fig.(a) Cn Note that except S0 = C0, each 0 fo 2fo 3fo 4fo 5fo 6fo (n-1) fo nfo spectral line in Fig. (a) at frequency f is replaced by the two spectral lines in Fig. (b), each with half amplitude, Fig.(b) one at frequency f and one at |Sn| frequency - f. ••• •••-nfo -(n-1)fo ••• - 6fo0-5fo -4fo -3fo -2fo -fo 0 fo 2fo 3fo 4fo 5fo 6fo ••• (n-1) fo nfo p. 12
13. 13. Fourier Series : ExampleConsider a unitary square wave defined by The Bn coefficients are given by 1, 0 < t < 0.5 2 T0 2πnt Bn = ∫ 2 x(t ) =  T0 x(t ) sin dt T0 −2 T0 − 1, 0.5 < t < 1 = 2 ∫ x(t ) sin (2πnt )dt 1and periodically extended outside this interval. 0The average value is zero, so = 2 ∫ sin (2πnt )dt + 2 ∫ − sin (2πnt )dt 0.5 1 0 0.5 A0 = 0.  cos(2πnt ) cos(2πnt )  1 0.5 = 2 − +  Recall that 2 T0 2πnt   2πn 0 2πn 0.5  An = ∫ 2 x(t ) cos dt T0 T0 −2 T0 2 = (1 − cos nπ) πn = 2 ∫ x(t ) cos(2πnt )dt 1 0 which results in = 2 ∫ cos(2πnt )dt + 2 ∫ − cos(2πnt )dt 0.5 1  4 0 0.5  , n is odd Bn =  nπ  sin (2πnt ) sin (2πnt )  0.5 1 = 2 −  0,  n is even   2πn 0 2πn 0.5  =0 Thus all An coefficients are zero. p. 13
14. 14. Fourier Series : ExampleThe Fourier series of a square wave of unitary amplitude with odd symmetry istherefore 4 1 1 x (t ) = (sin 2 πt + sin 6 πt + sin 10 πt + K) π 3 5 1st term 1st + 2nd terms 1st + 2nd + 3rd terms Sum up to the 6th term p. 14
15. 15. Fourier TransformRepresentation of an Aperiodic FunctionConsider an aperiodic function f(t)To represent this function as a sum of exponential functions overthe entire interval (-∞, ∞), we construct a new periodic functionfT(t) with period T.By letting T→∞, lim f T (t ) = f (t ) (13) T →∞ p. 15
16. 16. Fourier TransformThe new function fT(t) can be represented by an exponentialFourier series, which is written as ∞ f T (t ) = ∑ Fn e jn ω 0 t , n = −∞ (14)where 1 T /2 (15) Fn = T ∫−T / 2 f T (t ) e − jn ω 0 t dtand ω0 = 2π / T . p. 16
17. 17. Fourier TransformFor the sake of clear presentation, we set ∆ ∆ ω n = nω 0 , F ( ω n ) = TF n , (16)Thus, Eq.(14) and (15) become ∞ 1 f T (t ) = ∑T n = −∞ F ( ω n ) e jω n t , (17) T /2 (18) F (ω n ) = ∫−T / 2 f T (t ) e − jω n t dt .The spacing between adjacent lines in the line stream of fT(t)is ∆ω = 2π / T . (19) p. 17
18. 18. Fourier TransformUsing this relation for T, we get ∞ ∆ω f T (t ) = ∑ n = −∞ F (ω n )e jω n t 2π . (20)As T becomes very large, ∆ω becomes smaller and the spectrumbecomes denser.In the limit T → ∞, the discrete lines in the spectrum of fT(t) mergeand the frequency spectrum becomes continuous.Therefore, 1 ∞ lim f T (t ) = lim T →∞ T →∞ 2π ∑ n = −∞ F ( ω n ) e jω n t ∆ ω (21)becomes 1 ∞ 2 π ∫− ∞ f (t ) = F ( ω ) e jω t d ω (22) p. 18
19. 19. Fourier TransformIn a similar way, Eq. (18) becomes ∞ F (ω) = ∫−∞ f (t ) e − jω t dt . (23)Eq. (22) and (23) are commonly referred to as theFourier transform pair.Fourier Transform ∞ F (ω ) = ∫ −∞ f (t ) e − jω t dtInverse Fourier Transform 1 ∞ 2 π ∫− ∞ f (t ) = F ( ω ) e jω t d ω p. 19
20. 20. Spectral Density FunctionF(ω): The spectral density function of f(t). Fig. 3.2 A unit gate function Its spectral density graph sin( ω / 2 ) Sa ( ω / 2 ) = ω/2 p. 20
21. 21. Parseval’s TheoremThe energy delivered to a 1-ohm resistor is ∞ ∞ E= ∫ f (t ) dt = ∫ (24) 2 f (t ) f * (t ) dt . −∞ −∞Using Eq. (22) in (24), we get ∞ 1 ∞ *  1 ∞ E = ∫ f (t )  ∫ F (ω)e − jωt dω dt f (t ) = ∫− ∞ F (ω)e d ω jω t −∞  2π − ∞  2π 1 ∞ *  ∞ F (ω) ∫ f (t )e − jωt dt  dω 2π ∫−∞ =  −∞    1 ∞ * (25) = 2π ∫−∞ F (ω) F (ω)dω.Parseval’s Theorem: ∞ 1 ∞ ∫ 2 π ∫− ∞ 2 2 −∞ f (t ) dt = F ( ω) d ω. (26) p. 21
22. 22. Fourier Transform: Impulse FunctionThe unit impulse function satisfies ∞ ∫ δ( x)dx = 1, (27) −∞ ∞ x = 0, δ ( x) =  (28) 0 x ≠ 0.Using the integral properties of the impulse function, the Fouriertransform of a unit impulse, δ(t), is ∞ ℑ{δ(t )} = ∫ δ(t )e − jωt dt = e j 0 = 1. (29) −∞If the impulse is time-shifted, we have ∞ ℑ{δ(t − t0 )} = ∫ δ(t − t0 )e − jωt dt = e − jωt0 . (30) −∞ p. 22
23. 23. Fourier Transform: Complex Exponential Function ± jω tThe spectral density of e 0 will be concentrated at ±ω0. 1 ∞ ℑ {δ ( ω m ω 0 )} = 2 π ∫− ∞ −1 δ ( ω m ω 0 ) e jω t d ω 1 ± jω 0 t (31) = e , 2πTaking the Fourier transform of both sides, we have (32) ℑℑ −1 2π { {δ ( ω m ω 0 ) } = 1 ℑ e ± j ω 0 t }which gives { } ℑ e ± j ω 0 t = 2πδ (ω m ω 0 ) (33) p. 23
24. 24. Fourier Transform: Sinusoidal FunctionThe sinusoidal signals cos ω0tand sin ωcan be written in terms of 0tthe complex exponentials.Their Fourier transforms are given by {ℑ{cos ω 0 t } = ℑ 1 e jω 0 t + 1 e − jω 0 t 2 2 } = πδ ( ω − ω 0 ) + πδ ( ω + ω 0 ), (34) ℑ{sin ω0t} = ℑ {1 2j e jω0t − 21j e − jω0t } πδ(ω − ω0 ) − πδ(ω + ω0 ) = . j (35) p. 24
25. 25. Fourier Transform: Periodic FunctionsWe can express a function f(t) that is periodic with period T by itsexponential Fourier series ∞ f T (t ) = ∑ Fn e jn ω 0 t n = −∞ where ω0 = 2π/T. (36)Taking the Fourier transform, we have  ∞ jnω0 t  ℑ{ fT (t )} = ℑ ∑ Fn e  e.g. n = −∞  ∑ F ℑ{e } ∞ jnω0t = n A unit gate function Its Fourier transform n = −∞ ∞ = 2π ∑ Fn δ(ω − nω0 ). n = −∞ (37) Line spectrum of f(t) Its spectral density graph with period T p. 25
26. 26. Time and Spectral Density Functions p. 26
27. 27. Selected Fourier Transform Pairs p. 27
28. 28. Properties of Fourier TransformLinearity (Superposition) Time Shifting (Delay) a1 f1 (t ) + a 2 f 2 (t ) ↔ a1 F1 ( ω ) + a 2 F2 ( ω ) f (t − t 0 ) ↔ F (ω ) e − jω t 0Complex Conjugate Frequency Shifting (Modulation) f * (t ) ↔ F * (−ω) f ( t ) e jω 0 t ↔ F ( ω − ω 0 )Duality Convolution F (t ) ↔ 2 π f ( − ω ). f1 (t ) ∗ f 2 (t ) ↔ F1 ( ω ) F2 ( ω )Scaling 1  ω Multiplication f (at ) ↔ F  for a ≠ 0. a a f1 (t ) f 2 (t ) ↔ F1 ( ω ) ∗ F2 ( ω )Differentiation dn f (t ) ↔ ( jω) n F (ω) dt n p. 28
29. 29. Properties of Fourier TransformDuality F (t ) ↔ 2 π f ( − ω).Scaling 1 ω f ( at ) ↔ F  for a ≠ 0. a a p. 29
30. 30. Properties of Fourier TransformFrequency Shifting (Modulation) jω 0 t f (t ) e ↔ F (ω − ω 0 ) p. 30
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