TELE3113 Analogue & DigitalCommunications     Review of Fourier Transform                                   p. 1
Signal Representations(t) = A sin(2π fo t +φo ) or A sin(ωo t +φo )                                            Time-domain...
Energy and Power of SignalsFor an arbitrary signal f(t), the total energy normalized to unitresistance is defined as      ...
Periodic SignalA signal f(t) is periodic if and only if        f (t + T0 ) = f (t )     for all t   (*)where the constant ...
Deterministic & Random SignalsDeterministic signal can be modeled as a completely specifiedfunction of time.Example       ...
SystemMathematically, a system is a rule used for assigning a function g(t)(the output) to a function f(t) (the input); th...
Linear SystemIf a system is linear then superposition applies; that is, if            g1(t) = h{ f1(t) }, and g2(t) = h{ f...
Time-Invariant and Time-VaryingA system is time-invariant if a time shift in the input resultsin a corresponding time shif...
Fourier SeriesA periodic function of time s(t) with a fundamental period of T0 can berepresented as an infinite sum of sin...
Fourier SeriesAn alternative form of representing the Fourier series is                            ∞                      ...
Fourier SeriesThe exponential form of the Fourier series is used extensively incommunication theory. This form is given by...
Fourier Series                                                                                                    Amplitud...
Fourier Series : ExampleConsider a unitary square wave defined by                             The Bn coefficients are give...
Fourier Series : ExampleThe Fourier series of a square wave of unitary amplitude with odd symmetry istherefore            ...
Fourier TransformRepresentation of an Aperiodic FunctionConsider an aperiodic function f(t)To represent this function as a...
Fourier TransformThe new function fT(t) can be represented by an exponentialFourier series, which is written as           ...
Fourier TransformFor the sake of clear presentation, we set                    ∆                              ∆           ...
Fourier TransformUsing this relation for T, we get                         ∞                                              ...
Fourier TransformIn a similar way, Eq. (18) becomes                              ∞                F (ω) =   ∫−∞           ...
Spectral Density FunctionF(ω): The spectral density function of f(t).                      Fig. 3.2    A unit gate functio...
Parseval’s TheoremThe energy delivered to a 1-ohm resistor is             ∞                         ∞    E=   ∫       f (t...
Fourier Transform: Impulse FunctionThe unit impulse function satisfies          ∞      ∫       δ( x)dx = 1,               ...
Fourier Transform: Complex       Exponential Function                         ± jω tThe spectral density of e 0 will be co...
Fourier Transform: Sinusoidal FunctionThe sinusoidal signals  cos ω0tand                            sin ωcan be written in...
Fourier Transform: Periodic FunctionsWe can express a function f(t) that is periodic with period T by itsexponential Fouri...
Time and Spectral Density Functions                                      p. 26
Selected Fourier Transform Pairs                                   p. 27
Properties of Fourier TransformLinearity (Superposition)                                  Time Shifting (Delay)  a1 f1 (t ...
Properties of Fourier TransformDuality     F (t ) ↔ 2 π f ( − ω).Scaling                 1 ω             f ( at ) ↔  F ...
Properties of Fourier TransformFrequency Shifting (Modulation)               jω 0 t    f (t ) e            ↔ F (ω − ω 0 ) ...
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  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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