Figure 3.1 Effect on frequency spectrum of increasing period T 0. Consider the amplitude spectrum of a periodic waveform as shown in Figure 3.2.

Transcription

1 3. Fourier ransorm From Fourier Series to Fourier ransorm [, 2] In communication systems, we oten deal with non-periodic signals. An extension o the time-requency relationship to a non-periodic signal s(t) requires the introduction o the Fourier Integral. A nonperiodic signal can be viewed as a limiting case o a periodic signal, where the period approaches ininity. As approaches ininity, the periodic signal will eventually become a single non-periodic signal. his is shown in Figure 3.. Figure 3. Eect on requency spectrum o increasing period. he normalised energy o the non-periodic signal becomes inite and its normalised power tends to zero. Consider the amplitude spectrum o a periodic waveorm as shown in Figure 3.2. Figure 3.2 Amplitude spectrum o a periodic time unction. Let ω n = nω and ω = ω n+ - ω n = 2π. he Fourier series o a periodic waveorm s(t) with period can be written as s(t) = c n e jnω t = c n = 2π n e jnω t ω (3.) n = and a + /2 c n = s(t) e -jnω t dt (3.2) a /2 he constant a is usually set to. I approaches ininity, ω goes to. he harmonics get closer and closer together. In the limit, the Fourier series summation representation o s(t) becomes an integral, c n becomes a continuous unction S(ω), and we have a continuous requency spectrum. In summary, as ->, becomes, ω n becomes ω, and ω becomes dω. We have 3.

2 s(t) = 2π c n e jωt dω (3.3) and S(ω) = F[s(t)] = lim c n = s(t) e -jωt dt (3.4) It is also very common to work in terms o requency, = ω/2π, because spectrum analysers are usually calibrated in hertz. hus, we can express (3.3) and (3.4) as s(t) = F - [S()] = S() e j2πt d (3.5) and S() = F[s(t)] = s(t) e -j2πt dt (3.6) he unctions s(t) and S() are said to constitute a Fourier transorm pair, where S() is the Fourier transorm o a time unction s(t), and s(t) is the Inverse Fourier transorm (IF) o a requency-domain unction S(). Shorthand notation expressed in terms o t and : s(t) <-> S() Shorthand notation expressed in terms o t and ω : s(t) <-> S(ω) All physical waveorms encountered in engineering practice are Fourier transormable. In general, S() is a complex unction o requency. In two-dimensional cartesian orm, S() can be expressed as S() = X() + jy() (3.7) In polar orm, S() can be expressed as where S() = S() e jθ() (3.8) 3.2

3 S() = X 2 ( ) +Y 2 ( ) and θ() = tan - Y ( ) X( ) (3.9) S() represents the amplitude spectrum and θ() represents the phase spectrum o s(t). Example 3. Find the spectrum o an exponential pulse s(t) = e t, t >, t <. S() = e -t e -j2πt dt S() = +j 2π S() = +( 2π ) 2 θ() = -tan - (2π ) ransorms o Some Useul Functions [3]. Dirac Delta ime Function F[δ(t)] = δ(t) e -j 2πt dt = δ(t) <-> Also, it can be shown that δ(t - t ) <-> e -j 2π t 2. Dirac Delta Frequency-Domain Function F - [δ()] = δ() e j 2πt d = <-> δ() Also, it can be shown that e j 2π t <-> δ( - ) 3.3

4 Example 3.2 Find the spectrum o a sinusoid v(t) = A sin 2π t = A( ej 2π t e j 2π t 2j ). Since e j 2π t <-> δ( - ), we have V() = A 2j δ( - ) - A 2j δ( + ) V() = - A 2 j[δ( - ) - δ( + )] Figure 3.3 Spectrum o the periodic unction v(t) = A sin 2π t. 3. Rectangular, sin x/x, and riangular Pulses Observations: Figure 3.4 Spectra o (a) rectangular, (b) sin x/x, and (c) triangular pulses.. Figure 3.4a - Spectrum spreads out as the pulse width decreases. Bandwidth B = / Hz and S() decreases as /. 2. Figure 3.4c - Spectum spreads out as the pulse width decreases. Bandwidth B = / Hz and S() decreases as / 2. he smoother the time-domain unction, the more rapidly the spectrum decreases with increasing requency, packing more requency contents into a speciied bandwidth. An inverse time-bandwidth relation always exists. Bandwidth plays a signiicant role in determining transmission rate. 3.4

5 Function s(t) S() Rectangular Π t riangular Λ t ( ) [ ( ) [ sin π π sin π π Unit step, t > u(t) =, t < Signum sgn (t)=, t >, t < jπ Constant δ() Impulse at t= t δ(t - t ) e -j 2π t Sinc sin 2π Bt 2π Bt 2B Π 2B Phasor e j (2π t+ϕ) e j ϕ δ( - ) Cosine Sine Exponential, one sided cos(2π c t+ϕ) sin(2π c t+ϕ) e t /, t >, t < ] ] 2 2 δ() + j 2π, B denotes bandwidth 2 [ej ϕ δ( - c ) + e -jϕ δ( + c )] [e j ϕ δ( - 2j c ) - e -jϕ δ( + c )] +j 2π Exponential, two sided e - t / 2 +( 2π ) 2 Impulse train δ(t- k ) n δ( - n ), where = / k = = able 3.a Some Fourier transorm pairs expressed in terms o t and. 3.5

6 Function s(t) S(ω) Rectangular Π( t ) sin ω /2 ω /2 riangular Λ( t ) [ sin ω /2 ] ω /2, t > Unit step u(t) = πδ(ω) +, t < jω Signum sgn (t)=, t > 2, t < jω Constant 2πδ(ω) Impulse at t= t δ(t - t ) e -j ω t Sinc sin 2π Bt 2π Bt 2B Π ( ω 4πB ), B denotes bandwidth Phasor e j (ω t+ϕ) 2π e j ϕ δ(ω - ω ) Cosine cos(ω c t+ϕ) π[e j ϕ δ(ω - ω c ) + e-jϕ δ(ω + ω c )] Sine Exponential, one sided sin(ω c t+ϕ) e t /, t >, t < π j +jω [e j ϕ δ(ω - ω c ) - e-jϕ δ(ω + ω c )] Exponential, two sided e - t / 2 +(ω ) 2 Impulse train δ(t- k ) ω δ(ω - n ω k = ), n = where ω = 2π/ able 3.b Some Fourier transorm pairs expressed in terms o t and ω. 3.6

7 Properties o Fourier ransorms [3-5]. Symmetry (Duality) Property Proo. S(t) <-> s(-) s(t) = S() e j2πt d s(t) = S(x) e j2πxt dx s(-t) = S(x) e -j2πxt dx s(-) = S(x) e -j2πx dx s(-) = S(t) e -j2πt dt Hence, S(t) <-> s(-). 2. Scaling Property Proo. s(at) <-> a S( a ) F[s(at)] = s(at) e -j2πt dt Let t = at and a >, we get F[s(t )] = (/a) s(t ) e -j2π(/a)t dt F[s(t )] = (/a) S(/a) For a <, we get F[s(t )] = (-/a) s(t ) e -j2π(/a)t dt 3.7

8 F[s(t )] = (-/a) S(/a) Hence, s(at) <-> a S( a ). 3. ime Shiting (ime Delay) Property s(t - d ) <-> S()e -j 2π d Proo. Let α = t - d, dα = dt. F[s(t - d )] = s(t - d ) e -j 2πt dt = s(α) e -j 2π (α + d ) dα = e -j 2π d s(α) e -j 2π α dα = e -j 2π d S() Hence, s(t - d ) <-> S()e -j 2π d. 4. Frequency Shiting Property s(t)e j2π c t <-> S(- c ) Proo. S() = s(t) e -j2πt dt S( - c ) = s(t) e -j 2π(- c )t dt = [s(t) e j 2π c t ] e -j2πt dt 3.8

9 Hence, s(t)e j2π c t <-> S(- c ). 5. Dierentiation Property Proo. d n s (t ) dt n <-> (j2π) n S() Direct dierentation o the inverse Fourier transorm s(t) = F - [S()] = S() e j2πt d with respect to time n times. ds / dt = j2π S() e j2πt d ds / dt <-> j2π S() d 2 s / dt 2 = (j2π ) 2 S() e j2πt d d 2 s / dt 2 <-> (j2π ) 2 S() : d n s (t ) dt n = (j2π) n S() e j2πt d Hence, d n s (t ) dt n <-> (j2π) n S(). Dierentiation increases the high-requency content o a signal. he derivative o an even unction must be odd. Hence, the Fourier transorm o the derivative o the unction must be odd and imaginary. 6. Convolution Property s (t) * s 2 (t) <-> S ()S 2 () Proo. F[s (t) * s 2 (t)] = [ s (λ) s 2 (t - λ) dλ ] e -j2πt dt 3.9

10 F[s (t) * s 2 (t)] = s (λ)[ s 2 (t - λ) e -j2πt dt] dλ Since s(t - d ) <-> S() e -j 2π d (time shiting property), the inner integral is the Fourier transorm o s 2 (t - λ). We can write F[s (t) * s 2 (t)] = s (λ)s 2 ()e -j2πλ dλ F[s (t) * s 2 (t)] = S 2 () s (λ)e -j2πλ dλ F[s (t) * s 2 (t)] = S ()S 2 () hereore, s (t) * s 2 (t) <-> S ()S 2 () 7. Integration Property [4] t s(λ) dλ <-> j 2π S() + 2 S() δ() Proo. Because u(t - λ) =,, λ t, it ollows that λ > t t s(t) * u(t) = s(λ) u(t-λ) dλ = s(λ) dλ where u (t) is a unit step unction and the Fourier transorm o u (t) is U () = 2 δ () +. It ollows rom the time convolution property that j 2π s ( t) * u ( t) <-> S ( ) U ( ) and S ( ) U ( ) = S ( ) [ 2 δ() + ] = j 2π S() + S()δ(). Hence we have j 2π 2 t s(λ) dλ <-> j 2π S() + 2 S() δ(). 3.

11 Operation Function Fourier ransorm Linearity a s (t) + a 2 s 2 (t) a S () + a 2 S 2 () Conjugation s*(t) S*(-) Symmetry S(t) s(-) Scaling s(at) S( a a ) ime reversal s(-t) S(-) ime shit (delay) s(t - d ) S() e -j 2π d Frequency shit s(t)e j2π c t S(- c ) Real signal s(t) cos(2π c t + θ) 2 [ejθ S(- c ) +e -jθ S(+ c )] requency translation Bandpass signal Re{g(t)e j2π c t } 2 [G(- c ) + G*(-- c )] d n s (t ) Dierentiation dt n (j2π) n S() t Integration s(λ) dλ S() + S() δ() j 2π 2 Convolution s (t) * s 2 (t) = S ()S 2 () s (λ) s 2 (t-λ) dλ Multiplication s (t)s 2 (t) S () * S 2 () = S (λ) S 2 (-λ) dλ able 3.2a Some Fourier transorm properties expressed in terms o t and. 3.

12 Operation Function Fourier ransorm Linearity a s (t) + a 2 s 2 (t) a S (ω) + a 2 S 2 (ω) Conjugation s*(t) S*(-ω) Symmetry S(t) 2πs(-ω) Scaling s(at) S( ω a a ) ime reversal s(-t) S(-ω) ime shit (delay) s(t - d ) S(ω) e -jω d Frequency shit s(t)e j ω c t Real signal s(t) cos(ω c t + θ) requency translation S(ω-ω c ) 2 [ejθ S(ω-ω c )+e -jθ S(ω+ω c )] Bandpass signal Re{g(t)e jω c t } 2 [G(ω - ω c ) + G*(-ω - ω c )] d n s (t ) Dierentiation dt n (jω) n S(ω) t Integration s(λ) dλ S(ω) + πs() δ(ω) jω Convolution s (t) * s 2 (t) = S (ω)s 2 (ω) Multiplication s (λ) s 2 (t-λ) dλ s (t)s 2 (t) 2π S (ω) * S 2 (ω) = 2π S (λ) S 2 (ω -λ) dλ able 3.2b Some Fourier transorm properties expressed in terms o t and ω. Example 3.3 Use the scaling and real-signal requency-translation properties to ind the e t / sin ω t t > > Fourier transorm o a damped sinusoid s(t) =,,., t < From Example 3. we have 3.2

13 e t, t >, t < <-> +j 2π Using the scaling property with a = /, we get e t /, t >, t < <-> +j 2π Using the real-signal requency-translation property with θ = -π/2, we get S() = 2 [e-jπ/2 + j2π( ) +e jπ/2 ] + j2π( + ) he sin ω t actor causes the spectrum to move rom = to = I s(t) is real, then S(-) = S*() (3.) Proo. S (-) = s(t) e j 2 π t dt and S *() = F [ s ( t )]* = [ s(t) e -j2 π t dt]* = s*(t) e j2 π t dt. Because s(t) is real, s*(t) = s(t) and S(-) = S*(). 9. I s(t) is real, then and S(-) = S() (3.) θ(-) = -θ() (3.2) Proo. S (-) = S (-) e jθ (-) and S*() = S () e -jθ (). Because s(t) is real, S(-) = S*() and we see that (3.) and (3.2) are true. S() can be complex even though s(t) is real. 3.3

14 I s(t) is a: hen S() is a: Real and even unction o t Real and even unction o Real and odd Imaginary and odd Imaginary and even Imaginary and even Imaginary and odd Real and odd Complex and even Complex and even Complex and odd Complex and odd Example 3.4 able 3.3 Fourier transorm properties or various orms o s(t) [2]. cos 2π c t = ej 2π c t +e j 2π c t 2 sin 2π c t = ej 2π c t e j 2π c t 2j <-> 2 [δ(- c ) +δ(+ c )] <-> 2j [δ(- c ) -δ(+ c )] Figure 3.5 Fourier transorm spectrum o (a) cos 2π c t, and (b) sin 2π c t. Observations:. Figure 3.5a - A real and even unction in t gives a real and even unction in. 2. Figure 3.5b - A real and odd unction in t gives an imaginary and odd unction in. Parseval s heorem or the Fourier ransorm and Energy Spectral Density [4, 5] Parseval s heorem or the Fourier transorm states that i s (t) and s 2 (t) are two complex energy signals, then s (t) s* 2 (t) dt = S () S* 2 () d (3.3) 3.4

15 Proo. s (t) s* 2 (t) dt = [ S () e j2πt d] s* 2 (t) dt = S () s* 2 (t) e j2πt d dt Interchanging the order o integration, we have s (t) s* 2 (t) dt = S ()[ s 2 (t)e -j2πt dt]* d = S ()[F[s 2 (t)]]* d = S () S* 2 () d I s (t) = s 2 (t), then Rayleigh s energy theorem states that the normalised energy is E = s (t) 2 dt = S () 2 d (3.4) he energy spectral density (ESD) is deined or energy waveorms by E () = S () 2 (3.5) Proo. he auto-correlation o a complex energy waveorm s (t) is R (τ) = s* (t) s (t+ τ) dt R () = s* (t) s (t) dt = s (t) 2 dt = E = S () 2 d and the Fourier transorm o R (τ) is 3.5

16 E () =F[R (τ)] = R (τ) e -j2πτ dτ R (τ) =F - [E ()] = E () e j2πτ d R () = E ()d Hence E = E ()d and E () = S () 2. Power Spectral Density and Wiener-Khintchine heorem [3-5] he power spectral density and related concepts or a power waveorm s 2 (t) can be readily understood by deining a truncated waveorm s (t) as s (t) = s2(), t / 2< t < / 2, elsewhere = s 2 (t)π( t ) (3.6) and s (t) is an energy waveorm as long as is inite. Let P 22 () be the power spectral density o a power waveorm s 2 (t). he Wiener- Khintchine theorem states that the power spectral density and the autocorrelation unction are Fourier transorm pairs. R 22 (τ) <-> P 22 () (3.7) Furthermore, the average normalised power is P = <s (t)> = S 2rms = P 22 () d = R 22 () Proo. he average normalised power o a complex power waveorm s 2 (t) is /2 P = lim s 2 (t)s* 2 (t)dt /2 3.6

17 P = lim E s (t)s* (t)dt = lim where E is the normalised energy o the truncated waveorm s (t). he auto-correlation o a complex power waveorm s 2 (t) is /2 R (τ) = < s* 22 2 (t) s 2 (t+ τ)> = lim s* 2 (t) s 2 (t+τ)dt /2 /2 R () = lim 22 s* 2 (t) s 2 (t) dt /2 /2 = lim s 2 (t) 2 dt /2 = P and the Fourier transorm o R 22 (τ) is P 22 () =F[R 22 (τ)] = R 22 (τ) e -j2πτ dτ R (τ) =F - [P ()] = P () e j2πτ d R () = P ()d Hence P = P 22 ()d. he average normalised power o a power waveorm is now related to the power spectral density. he power spectral density (PSD) or a power waveorm s 2 (t) is S ( ) 2 P () = lim 22 (3.8) where S () is the Fourier transorm o the truncated waveorm s (t). 3.7

18 Proo. R (τ) <-> E () = S () 2 and R 22 (τ) <-> P 22 () he auto-correlation o a complex power waveorm s* 2 (t) is /2 R (τ) = < s* 22 2 (t) s 2 (t+ τ)> = lim s* 2 (t) s 2 (t+τ)dt /2 R (τ ) R (τ) = lim 22 s* (t) s (t+τ)dt = lim Hence we have P 22 () = E ( ) lim S ( ) 2 = lim. he power spectral density is always a real nonnegative unction o requency. It is not sensitive to the phase spectrum o the truncated waveorm s (t). hus, A sin 2π t and A cos 2π t have the same PSD because the phase has no eect on the power spectral density. 3.8

19 Energy Signal Power Signal /2 E = s (t) s* (t) dt P = lim s 2 (t)s* 2 (t)dt /2 E = S () S* () d = lim /2 R (τ) = s* (t)s (t+τ)dt R (τ) = lim 22 s* 2 (t)s 2 (t+τ)dt /2 R (τ ) = lim E () = S () 2 E ( ) S ( ) 2 P () = lim = lim 22 R (τ) <-> E () R (τ) <-> P () E = E () d P = P 22 () d able 3.4 Relationships or energy and power signals. Fourier ransorm o Periodic Signals [] So ar we have used the Fourier series and the Fourier transorm to represent periodic and nonperiodic signals, respectively. For periodic signals, we can use an impulse unction in the requency domain to represent discrete components o periodic signals using Fourier transorms. With this approach, both periodic and nonperiodic signals can be incorporated in a common Fourier-transorm ramework. Recall: Aδ(t) <-> A Aδ(t- t ) <-> Ae -j2πt A <-> Aδ() Ae j2π t <-> Aδ(- ) 3.9

20 he complex Fourier series o a periodic signal is given by s(t) = n = c n e j2πn t and the Fourier transorm o s(t) is S() = c n δ(- n ) n = Example 3.5 he complex Fourier series o a periodic rectangular waveorm s(t) is s(t) = c n e j2πn t <-> n = c n δ(- n ) n = where c n = A m τ( sin 2π n τ /2 ). 2π n τ /2 Figure 3.6 (a) A periodic rectangular waveorm s(t), and (b) the Fourier transorm spectrum o s(t). Example 3.6 A periodic impulse s(t) is s(t) = k = δ(t- k ) <-> n δ( - n ), where = /. = Figure 3.7 (a) Periodic impulse s(t), and (b) Fourier transorm spectrum o s(t). Summary For periodic signals, the impulse unction δ() provides a uniied method o describing such signals in the requency domain using the Fourier transorm.. Express the periodic signal in a complex Fourier series. 2. ake the Fourier transorm. I a signal s(t) = s p (t)s a (t), where s p (t) and s a (t) are the corresponding periodic 3.2

21 and nonperiodic components, respectively, we have s(t) = s p (t)s a (t) <->S p () * S a () I a signal s(t) = s p (t) + s a (t), we have s(t) = s p (t) + s a (t) <->S p () + S a () In using the impulse unction in the requency domain, we must bear in mind that we are dealing with signals having ininite or undeined energy, and the concept o energy spectral density no longer exists. Reerences [] M. Schwartz, Inormation ransmission, Modulation, and Noise, 4/e, McGraw- Hill, 99. [2] J. D. Gibson, Modern Digital and Analog Communications, 2/e, Macmillan Publishing Company, 993. [3] L. W. Couch II, Digital and Analog Communication Systems, 5/e, Prentice Hall, 997. [4] B. P. Lathi, Modern Digital and Analog Communication Systems, 3/e, Oxord University Press, 998. [5] H. P. Hsu, Analog and Digital Communications, McGraw-Hill,

22 c n Am τ Envelope - s( t ) Am τ ime = 4 τ 2π First zero crossing ω 2π 4π 6π 2 ω τ τ τ ω c n Am τ s( t ) Am τ 2π - ime = 8τ ω 2π τ 4 ω 4π τ 6π τ c n Am τ ω s( t ) Am τ ime -> 2π 4π 6π τ τ τ ω Figure 3. Eect on requency spectrum o increasing period. 3.22

23 c n ω 2ω 2π ω =... ω n ω n + Figure 3.2 Amplitude spectrum o a periodic time unction. ω V() θ ( ) A /2 9 o o Figure 3.3 Spectrum o the periodic unction A sin 2π t. 3.23

24 ime domain Π( t ) = {, t < 2 Frequency domain sin π π, t > t (a) B sin 2π Bt 2 2π Bt 2B Π ( ) 2B - B - 2B Λ t ( ) 2B B t, t > - ={ t (b), t < -B B sin π ( π ) 2 - t (c) Figure 3.4 Spectra o (a) rectangular, (b) sin x/x, and (c) triangular pulses. 3.24

25 Amplitude Phase δ ( + ) δ ( - ) 2 c /2 2 c - c c π/2 - c c π/2 (a) Amplitude Phase j - δ ( + ) 2 c j /2 - c c j -j /2 - - δ ( - ) 2 c (b) π/2 - c c π/2... Figure 3.5 Fourier transorm spectrum o (a) cos 2π c t, and (b) sin 2π c t.... -n s( t ) A m ime τ τ (a) S () A sin m τ 2πn τ/2 2πn τ/ (b) 2 [ ] )... n... δ( - n Figure 3.6 (a) A periodic rectangular waveorm s(t), and (b) the Fourier transorm spectrum o s(t). 3.25

26 δ ( t + ) δ ( t ) δ ( t - ) t - (a) δ( +2 ) δ ( -2 ) δ( ) (b) Figure 3.7 (a) Periodic impulse s(t), and (b) Fourier transorm spectrum o s(t). 3.26