2.1 Introduction 2.22 The Fourier Transform 2.3 Properties of The Fourier Transform 2.4 The Inverse Relationship between Time and Frequency 2.

Transcription

1 Chapter2 Fourier Theory and Communication Signals Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University

2 Contents 2.1 Introduction 2.22 The Fourier Transform 2.3 Properties of The Fourier Transform 2.4 The Inverse Relationship between Time and Frequency 2.5 Dirac Delta Function 2.6 Fourier Transform of Periodic Signals 2.7 Transmission of Signals Through Linear Systems 2.8 Filters 2.9 Low-Pass and Band-Pass Signals 2.10 Band-Pass Systems 2.11 Phase and Group Delay 2.12 Sources of Information 2.13 Numerical Computation of the Fourier Transform 2

3 Chapter 2.1 Introduction Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University

4 Chapter 2.1 Introduction We identify deterministic signals as a class of signals whose waveforms are defined dexactly as functions of time. In this chapter we study the mathematical description of such signals using the Fourier transform that provides the link between the time-domain and frequency-domain descriptions of signal. Another related issue that we study in this chapter is the representation of linear time-invariant systems. Filters of different kinds and certain communication channels are important examples of this class of systems. 4

5 Chapter 2.2 The Fourier Transform Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University

6 Chapter 2.2 The Fourier Transform Fourier transform is defined as ( ( exp( 2π G f = g t j ft dt g(t denote a nonperiodic deterministic signal. j = 1 variable f denotes frequency and t denotes time.. Inverse Fourier transform is defined as ( ( exp( 2π g t = G f j ft df 6

7 Chapter 2.2 The Fourier Transform We have used a lowercase letter to denote the time function and a uppercase letter to denote the corresponding frequency function. The functions g(t and G( f are said to constitute a Fourier-transform pair. For the Fourier transform of a signal g(t to exist, it is sufficient, but not necessary, that g(t satisfies three sufficient conditions known collectively as Dirichlet s conditions: The function g(t is single-valued, with a finite number of maxima and minima ii in any finite fiit time interval. it The function g(t has finite number of discontinuities in any finite time interval. The function g(t is absolutely integrable, i.e. g( t dt 7 <

8 Chapter 2.2 The Fourier Transform We may safely ignore the question of the existence of the Fourier transform of a time function when it is an accurately specified description of a physically realizable signal. Physical realizability is a sufficient condition for the existence of a Fourier transform. All energy signals are Fourier transformable. Energy signals: g g ( t 2 dt < ; ( P= 0 1 T P = lim t 2 dt g ( T 2T T Power Signal:0 < P < ( E = Plancherel s theorem: if a time function g(t is such that the value of the energy g ( t 2 dt < is defined and finite, then the Fourier transform G( f of the function g(t exists and 2 A lim g( t G( f exp( j2 π ft df dt 0. A = A 8

9 Chapter 2.2 The Fourier Transform Notations time t measured in second (s frequency f measured in Hertz (Hz angular frequency ω = 2π f (radians per second, rad/s. A convenient shorthand notation for the transform relations: Fourier transformation ti ( F g ( t G f = Inverse Fourier transformation 1 ( t F G ( f g t = F G f where and play the roles of linear operators. F [ ] F 1 [ ] Fourier-transform pair g ( t G( f 9

10 Chapter 2.2 The Fourier Transform Continuous Spectrum By using the Fourier transform operation, a pulse signal g(t of finite energy is expressed as a continuous sum of exponential functions with frequencies in the interval - ~. The amplitude of a component of frequency f is proportional to G( f, where G( f is the Fourier transform of g(t. At any frequency f, the exponential function exp(j2 πft is weighted by the factor G( f df, which is the contribution of G( f i in an infinitesimal ifiit i lit interval ldf centered at tthe frequency f. We may express the function g(t in terms of the continuous sum of such infinitesimal components: ( ( exp ( 2 π g t = G f j ft df 10

11 Chapter 2.2 The Fourier Transform In general, the Fourier transform G( f is a complex function of frequency f : G( f = G ( f exp jθ ( f ( f is called the continuous amplitude spectrum of g ( t ( f is called the continuous phase spectrum of g( t G f θ If g(t is a real-valued function of t, then G(-f =G*( f G(-f = G( f : an even function of f Θ(-f f =-Θ( f : an odd function of f j2 ( ( π ft G f g t e dt * j 2π ft *( = ( ( = G f g t e dt = ( g t e ( f = G The spectrum of a real-valued signal exhibits conjugate symmetry. j2π ft dt 11

12 Chapter 2.2 The Fourier Transform [Example 2.1] Rectangular Pulse Define a rectangular function of unit amplitude and unit duration: 1 1 1, < t < 2 2 rect ( t Real-Valued and Symmetric Real-Valued and Symmetric = 0, t 1 2 t t g( t = A rect A rect AT sinc( ft T T (2.10 T /2 G f = Aexp j2π ft dt ( ( T /2 T /2 T /2 ( cos(2 π sin(2 π = A ft j ft dt ( π ft ( π ft T /2 sin 2 = 2A cos(2 π ft dt = 2A 0 2π f sin = AT ATsinc ft π ft 12 ( T 2 0 sinc ( λ sin ( πλ πλ

13 Chapter 2.2 The Fourier Transform sinc function ( sinc λ sin ( πλ πλ As the pulse duration T is decreased, the first zero-crossing of the amplitude spectrum G( f moves up in frequency. The relationship between the time-domain and frequency-domain is an inverse one. A pulse, narrow in time, has a significant frequency description over a wide range of frequencies, and vice versa. 13

14 Chapter 2.2 The Fourier Transform [Example 2.2] Exponential Pulse At truncated t dform of a decaying exponential pulse is shown in the following figure (adecaying exponential pulse. (brising exponential pulse. 14

15 Chapter 2.2 The Fourier Transform [Example 2.2] Exponential Pulse(cont. It is convenient to mathematically ti define the decaying exponential pulse using the unit step function. An unit step function is defined as: ( u t 1, t > 0 1 =, t = t < 0 Decaying exponential pulse of ffi figure (a can be expressed as gt ( = exp( atut ( 15

16 Chapter 2.2 The Fourier Transform the Fourier transform of this pulse is ( = ( ( 2π G f exp at exp j 2 ft dt 0 exp ep t ( a+ j2 π f exp = t( a j2 π f dt 0 + = a+ j2π f 1 = a + j2π f ( 0 the Fourier-transform pair for the decaying exponential pulse of figure (a is therefore exp( ( at u ( t 1 a + j2π f 16

17 Chapter 2.2 The Fourier Transform [Example 2.2] Exponential Pulse(cont. Rising i exponential pulse of Fig. (b ( = exp( ( exp( ( at u ( t g t at u t ( 0 exp ( exp ( 2 π G f = at j ft dt ( π 0 1 = exp t a j2 f dt = a j2π f The decaying and rising exponential pulses are both asymmetric functions of time t. Their Fourier transforms are therefore complex valued. 1 a j 2π f Truncated decaying and rising exponential pulses have the same amplitude spectrum, but the phase spectrum of the one is the negative of that of the other. 17

18 Fourier-transform Pairs 18

19 Chapter 2.3 Properties of The Fourier Transform Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University

20 Chapter 2.3 Properties of the Fourier Transform Summary of properties of the Fourier transform Property Mathematical Description 1. Linearity ag ( t + bg ( t ag ( f + bg ( f 2. Time scalingg 3. Duality , a and b are constants. 1 f g ( at G a a a where is constant ( ( ( ( If g t G f then G t g f 4. Time shifting g( t t G( f exp( j2π ft 5. Frequency shifting 0 0 ( j π f t g( t G( f f exp 2 c c 6A 6. Area under g(t ( g( t dt = G( 0 20

21 Chapter 2.3 Properties of the Fourier Transform Property Mathematical Description 7. Area under G( f 8Diff 8.Differentiation i i in the time domain 9. Integration in the time domain 10. Conjugate functions 11. Multiplication in the time domain 12. Convolution in the time domain 13. Rayleigh s t g ( 0 ( = G f df d g t j 2π fg f dt ( ( ( 0 g 1 G ( τ d τ ( j2 f G f + δ π 2 f ( G then g( ( f g ( t G ( f If g t ( ( ( ( g t g t G λ G f λ dλ ( τ ( τ τ ( ( g g t d G f G f energy theorem ( = ( 21 2 g t dt G f df 2

22 Chapter 2.3 Properties of the Fourier Transform Property 1:Linearity (Superposition Lt Let g1 ( t G1 ( f and g2 ( t G2 ( f. Then for all constants t c 1 and c 2, we have ( + ( + ( cg t cg t cg cg f Proof: the proof of this property follows simply from the linearity of the integrals defining G( f and g(t. 22

23 Chapter 2.3 Properties of the Fourier Transform [Example 2.3] Combinations of Exponential Pulses Consider a double exponential pulse ( g t ( at exp, t > 0 = 1, t = 0 exp ( at, t < 0 =exp ( at Symmetric in time domain. Spectrum is real and symmetric. This pulse may be viewed as the sum of a truncated t decaying exponential pulse and a truncated rising exponential pulse a G( f = + = 2 a+ j2π f a j2π f a + ( 2π f 2 2a a exp( at 2 a + 2π f 23 ( 2

24 Chapter 2.3 Properties of the Fourier Transform [Example 2.3] Combinations of Exponential Pulses(cont. ( g t ( at exp, t > 0 = 0, t = 0 exp ( at, t < 0 + 1, t > 0 ( sgn t = 0, t = 0 1, t < 0 ( = exp( sgn ( g t a t t 24

25 Chapter 2.3 Properties of the Fourier Transform [Example 2.3] Combinations of Exponential Pulses(cont. 1 1 F exp( a t sgn ( t = a + j2π f a j2π f = a 2 j4π f + ( 2 π f 2 exp ( at sgn ( t a 2 j4π f + ( 2π f 2 The Fourier transform is odd and purely imaginary. In general, a real odd-symmetric time function has an odd and purely imaginary function as its Fourier transform. 25

26 Chapter 2.3 Properties of the Fourier Transform Property 2:Time Scaling Compression of a function in Proof: ( = ( exp( 2π the time domain is equivalent to the expansion of its Fourier transform in the frequency domain, or vice versa. 1 f Let gt ( G( f. Then g( at G a a τ F g at g at j ft dt τ = at t = a 1 f 1 f For a> 0 : F g( at = g( τ exp j2π τ dτ = G a a a a 1 f For a< 0 : F g( at = g( τ exp j2π τ dτ a a 1 f 1 f = g( τ exp j2π τ dτ G = a a a a 26 Q.E.D.

27 Chapter 2.3 Properties of the Fourier Transform Property 3:Duality ( ( G ( t g ( f If g t G f, then Proof ( j2 ( = ( π ft G f = g t e dt 2 π gt ( = G( f e j ft df 2 t f f t g( f = G( t e j π ft dt j2π ft g( f = Gte ( dt= FGt { ( } t rect sinc (2.10 T A f A f A sinc( 2 Wt rect = rect 2W 2W 2W 2W [Example 2.4] A AT ( ft 27

28 Chapter 2.3 Properties of the Fourier Transform Property 4:Time Shifting ( ( g( t t G( f exp( j2π ft If g t G f, then 0 0 Let t t 0 Proof : τ = ( ( = ( ( π F g t t0 g t t0 exp j2 ft dt ( 0 ( ( = exp j2π ft g τ exp j2π fτ dτ ( j π ft G ( f = exp j2 f f 0 The amplitude of G( f is unaffected by the time shift, but its phase is changed by the linear factor -2πft 0. 28

29 Chapter 2.3 Properties of the Fourier Transform Property 5:Frequency Shifting (Modulation Theorem ( ( ( ( ( If g t G f, then exp j2π fct g t G f fc where f is a real constant Proof: f c ( π ( = ( π ( F exp j2 fct g t g t exp j2 t f fc dt ( = G f f c Multiplication of a function by the factor exp(-2πf c t is equivalent to shifting its Fourier transform in the positive direction by the amount f c. 29

30 Chapter 2.3 Properties of the Fourier Transform [Example 2.5] Radio Frequency (RF Pulse ( Consider the pulse signal g t shown in figure (a which hconsists of a sinusoidal wave of amplitude A and frequency f c, extending in duration from t =-T/2 to t = T/2. This signal is sometimes referred to as an RF pulse when the frequency f c falls in the radio-frequency band. The signal g( t of figure (a may be expressed mathematically as follows t g t = A rect cos 2 fct T ( ( π 30

31 Chapter 2.3 Properties of the Fourier Transform [Example 2.5] Radio Frequency (RF Pulse (cont. we note that 1 cos( 2π ft c = exp( 2 exp( 2 2 j π ft c + j π ft c applying the frequency-shifting property to the Fourier-transform pair, we get the desired result AT G( f = { sinc ( sinc ( } 2 T f fc + T f + fc in the special case of f c T>>1, we may use the approximate result ( G f AT sinc T( f fc, f > 0 2 0, f = 0 AT sinc T( f + fc, f <

32 Chapter 2.3 Properties of the Fourier Transform [Example 2.5] Radio Frequency (RF Pulse (cont. The amplitude spectrum of the RF pulse is shown in figure (b. This diagram, in relation to figure in page 12, clearly illustrates the frequency-shifting property of the Fourier transform. 32

33 Chapter 2.3 Properties of the Fourier Transform Property 6:Area Under g(t G f g( t dt = G( 0 ( ( Physical meaning for G(0? If g t, then That is, the area under a function g(t is equal to the value of its Fourier-transform G( f at f =0. This result can be obtained by putting f =0 in the formula of Fourier transform. Property 7:Area Under G( f ( ( ( ( ( = ( exp( 2π G f g t j ft dt If g t G f, then g 0 = G f df That is, the value of a function g(tatt t =0 is equal to the area under its Fourier-transform G( f. This result can be obtained by putting t =0 in the formula of inverse Fourier transform. 33 ( ( exp( 2π g t = G f j ft df

34 Chapter 2.3 Properties of the Fourier Transform Property 8:Differentiation in the Time Domain ( ( Lt Let g t G f, and assume that thtthe first tderivative of g(t ( is Fourier transformable. Then d g ( t j 2 π f G ( f (2.31 dt That is, differentiation of a time function g(t (thas the effect of multiplying its Fourier transform G( f by the factor j2πf. If we assume that the Fourier transform of the higher-order derivative exists, then d n g ( t n ( j 2π f n G ( f dt Proof:This result is obtained by taking the first derivative of both sides of the integral defining the inverse Fourier transform. 34 ( ( exp( 2π g t = G f j ft df

35 Chapter 2.3 Properties of the Fourier Transform [Example 2.6] Gaussian Pulse We will derive the particular form of a pulse signal that t has the same mathematical form as its own Fourier transform. By differentiating the formula for the Fourier transform G( f with respect to f, we have d G ( f = t exp 2π ft dt j2π tg( t G( f g( ( j df d g t j f G f dt Add (2.31 ( 2 π ( plus j times j2π tg ( t G( f ( ( dg t dt dg t If = 2πtg ( t, then dt ( dg f + 2πtg ( t j + 2π fg f df ( dg f df ( ( = 2π fg f d df 35

36 Chapter 2.3 Properties of the Fourier Transform [Example 2.6] Gaussian Pulse (cont. Since the pulse signal g(t ( and dits Fourier transform G( f satisfy the same differential equation, they are the same function, i.e. G( f = g( f, whereg( f is obtained from g(t by substituting f for t. ( dg t Since 2π tg ( t, we can obtain dt = ( ( 2 g t = exp π t This pulse is called a Gaussian Pulse. ( 2 exp πt dt = 1 exp( π t 2 exp( π f 2 36

37 Chapter 2.3 Properties of the Fourier Transform Property 9:Integration in the Time Domain ( ( Lt Let g t G f. Then provided iddthtg(0 that G(0=0, 0 we have t 1 g ( τ d τ ( j2π f G f (2.39 d t g t = g τ dτ dt Proof: ( ( Applying the time-differentiation property of the Fourier transform d x ( t j2 π f X ( f dt { } t ( ( 2 ( F g t = G f = j π f F g τ d τ 37

38 Chapter 2.3 Properties of the Fourier Transform [Example 2.7] Triangular Pulse consider the doublet pulse g 1 (t shown in Fig. (a. By integrating this pulse with respect to time, we obtain the triangular pulse g 2 (t. The doublet pulse of figure (a is real and odd-symmetric and its Fourier transform is therefore odd and purely imaginary. The triangular pulse of figure (b is real and symmetric and its Fourier transform is therefore symmetric and purely real. 38

39 Chapter 2.3 Properties of the Fourier Transform [Example 2.7] Triangular Pulse (cont. g 1 (t consists of ftwo rectangular pulse: amplitude A, defined for the interval T t Fourier transform: AT sinc( ft exp( j π ft amplitude A, defined for the interval 0 t T Fourier transform: ATsinc( ft exp( jπ ft Invoking the linearity property of the Fourier transform of ( = sinc ( exp ( π exp ( π = 2 jat sinc( ft sin ( π ft G ( = 1 sin ( π ft ( = ( = sinc( G1 f AT sinc ft exp j ft exp j ft G2 f G1 f AT ft j2π f π f = AT 2 2 sinc ( ft 39 g1 ( t

40 Chapter 2.3 Properties of the Fourier Transform Property 10:Conjugate Functions ( ( If g t G f, then for a complex-valued l dtime function g(t ( we have g t G f ( ( Proof: ( ( exp( 2π g t = G f j ft df taking the complex conjugates of both sides yields ( ( exp( 2π g t = G f j ft df Replacing f with f gives ( ( exp ( 2 π ( exp ( 2 π g t = G f j ft df = G f j ft df Corollary: y g ( t G ( f 40

41 Chapter 2.3 Properties of the Fourier Transform [Example 2.8] Real and Imaginary Parts of a Time Function Re ( = Re ( + Im ( ( = Re ( Im ( g t g t j g t g t g t j g t j 1 Re g( t G( f + G ( f 2 1 Im g( t G( f G ( f 2 j g( t = g( t + g ( t Im g( t = g( t g ( t (Im[g(t]=0 If g(t is a real-valued time function, we have G( f = G*(- f. In other words, G( f exhibits conjugate symmetry. 41

42 Chapter 2.3 Properties of the Fourier Transform Property 11:Multiplication in the Time Domain ( ( ( ( Lt Let g t G f and g t G f.then 1 1 Proof:Let s denote 2 2 ( ( ( ( g t g t G λ G f λ dλ ( ( ( g t g t G f ( ( ( ( π G12 f = g1 t g2 t exp j2 ft dt For g (t, we have: ( ( ( exp 2π g t = G f j f t df ( = ( ( π ( ' ' ' ' ' ' G12 f g1 t G2 f exp j2 f f t df dt Define: λ = f f ' G12 ( f = G2 ( f λ g1 ( t exp( j2πλt dt dλ 42

43 Chapter 2.3 Properties of the Fourier Transform The inner integral is recognized as G 1 (λ 12 ( = 1 ( 2 ( G f G λ G f λ dλ Q.E.D. This integral is known as the convolution integral expressed in the frequency domain, and the function G 12 ( f is referred to as the convolution of G 1 ( f and G 2 ( f. The multiplication li of two signals in the time domain is transformed into the convolution of their individual Fourier transforms in the frequency domain. This property is known as the multiplication theorem. Notation: G f = G f G f 12 ( 1 ( 2 ( ( ( ( ( g t g t G f G f ( ( = ( ( G f G f G f G f

44 Chapter 2.3 Properties of the Fourier Transform Property 12:Convolution in the Time Domain ( ( ( ( Lt Let g t G f and g t G f, then Proof: ( τ ( τ τ ( ( g g t d G f G f j π ft g12( t = G1 ( f G2 ( f e df Let = ( ( λ = t u j2 uf j2 ft G π π 1( f g 2( ue due df 2 = g2( t G1( f e j π f λ λ df dλ ( ( = g λ g t λ dλ Q.E.D.

45 Chapter 2.3 Properties of the Fourier Transform We may thus state that the convolution of two signals in the time domain is transformed into the multiplication of their individual Fourier transforms in the frequency domain. This property p is known as the convolution theorem. Property 11 and property 12 are the dual of each other. Shorthand notation for convolution: ( ( ( ( g t g t G f G f

46 Chapter 2.3 Properties of the Fourier Transform Property 13:Rayleigh s Energy Theorem (Parseval'sor Plancharel stheorem Let g(t be defined over the entire interval -<t< and assume its Fourier transform G( f exists. If the energy of the signal satisfies then - ( 2 E = g t dt < ( = ( 2 2 g t dt G f df G( f 2 is defined as the energy spectral density (valid for energy signal. For power signal, we define power spectral density S( f : 1 T 46 2 ( = lim ( ( deterministic signal = P S f df T 2 g t dt T T

47 Chapter 2.3 Properties of the Fourier Transform Proof: 2 ( * ( ( E g t dt g t g t dt = = = = = - = - *( j2 ( ft g t π G f e df dt j2 ( *( ft G f π g t e dt df * 2 ( ( j ft G f π g t e dt df = ( *( G f G G( f 2 df f df 47

48 Chapter 2.3 Properties of the Fourier Transform [Example 2.9] Sinc Pulse (continued Consider the sinc pulse A sinc(2wt. The energy of fthis pulse equals 2 2 E = A sinc ( 2Wt dt The integral in the right-hand side of this equation is rather difficult to evaluate. From example 2.4, the Fourier transform of the sinc pulse A sinc(2wt is equal to (A/2Wrect( f /2W. Applying Rayleigh s energy theorem 2 A 2 f E = rect df 2W 2W 2 2 W = A A 2W df = W 2W 48

49 Chapter 2.3 Properties of the Fourier Transform Summary of Properties for Fourier Transform g(t Real Valued Symmetric Asymmetric Real Valued and Symmetric Real Valued and Odd Symmetric G( f Spectrum Conjugate Symmetry Real Valued Complex Valued Real Valued and Symmetric Odd and Purely Imaginary Compression of a function in the time domain is equivalent to the expansion of its Fourier transform in the frequency domain, or vice versa. 49

50 Chapter 2.4 The Inverse Relationship between Time and Frequency Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University

51 Chapter 2.4 The Inverse Relationship between Time and Frequency The time-domain and frequency-domain description of a signal are inversely related: If the time-domain description of a signal is changed, then the frequency-domain description of the signal is changed in an inverse manner, and vice versa. If a signal is strictly limited in frequency, then the time-domain description of the signal will trail on indefinitely. A signal is strictly limited in frequency or strictly band limited if its Fourier transform is exactly zero outside a finite band of frequencies. If a signal is strictly limited in time, then the spectrum of the signal lis infinite i in extent. A signal is strictly limited in time if the signal is exactly zero outside a finite time interval. 51

52 Chapter 2.4 The Inverse Relationship between Time and Frequency Bandwidth The bandwidth of a signal provides a measure of the extent of significant spectral content of the signal for positive frequencies. When the signal is strictly band limited, the bandwidth is well defined. When the signal is not strictly band-limited, there is no universally accepted definition of bandwidth. A signal is said to be low-pass if its significant spectral content is centered around the origin. A signal is said to be band-pass if its significant spectral content is centered around ±f c,wheref f c is a nonzero frequency. 52

53 Chapter 2.4 The Inverse Relationship between Time and Frequency Bandwidth (cont. When the spectrum of a signal is symmetric with a main lobe bounded by well-defined nulls(i.e. frequencies at which the spectrum is zero, we may use the main lobe as the basis for defining the bandwidth of the signal. When the signal is low-pass, the bandwidth is defined as one half the total width of the main spectral lobe, since only one half of this lobe lies inside id the positive frequency region. When the signal is band-pass with main spectral lobes centered around ±f c, where f c is large, the bandwidth is defined as the width of the main lobe for positive frequency. This definition of bandwidth is called the null-to-null bandwidth. 53

54 Chapter 2.4 The Inverse Relationship between Time and Frequency Bandwidth (cont. 54

55 Chapter 2.4 The Inverse Relationship between Time and Frequency 3-dB Bandwidth When the signal is low-pass, the 3-dB bandwidth is defined as the separation between zero frequency, where the amplitude spectrum attains its peak value, and the positive frequency at which the amplitude spectrum drops to 1/ 2 of its peak value. When the signal is band-pass, centered at ±f c, the 3-dB bandwidth is defined as the separation (along the positive frequency axis between the two frequencies at which the amplitude spectrum of the signal drops to 1/ 2 of the peak value at f c. Advantage : it can be read directly from a plot of the amplitude spectrum. Disadvantage: it may be misleading if the amplitude spectrum has slowly decreasing tails. 55

56 Chapter 2.4 The Inverse Relationship between Time and Frequency 3-dB Bandwidth 56

57 Chapter 2.4 The Inverse Relationship between Time and Frequency Root Mean Square (rms Bandwidth Root Mean Square (rms bandwidth, defined as the square root of the second moment of a properly normalized form of the squared amplitude spectrum of the signal about a suitably chosen point. The rms bandwidth of a low-pass signal is formally defined as: f G( f df W rms 2 = G( f df An attractive feature of the rms bandwidth is that it lends itself more readily to mathematical evaluation than the other two definitions of bandwidth, but it is not as easily measurable in the laboratory. 57

58 Chapter 2.4 The Inverse Relationship between Time and Frequency Time-Bandwidth product For any family of pulse signals (e.g. the exponential pulse that differ in time scale, the product of the signal s duration and its bandwidth is always a constant, t as shown by (duration (bandwidth = constant The product is called the time-bandwidth idth product or bandwidth-duration product. If the duration of a pulse signal is decreased by reducing the time scale by a factor a, the frequency scale of the signal ss spectrum, and therefore the bandwidth of the signal, is increased by the same factor a. 58

59 Chapter 2.4 The Inverse Relationship between Time and Frequency Time-Bandwidth product (cont. Consider the rms bandwidth. The corresponding definition iti for the rms duration is T t g( t dt rms 2 = g( t 2 dt The time-bandwidth idthproduct thas the following form: 1 T rmsw rms 4π Gaussian pulse satisfies this condition with the equality sign. 59

60 Chapter 2.5 Dirac Delta Function Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University

61 Chapter 2.5 Dirac Delta Function The Dirac delta function, denoted by δ(t, is defined as having zero amplitude everywhere except at t = 0, where it is infinitely large in such a way that it contains unit area under its curve; i.e. δ ( t = 0, t 0 δ ( tdt = 1 The delta function δ(tisaneven function of time t. Sifting property of the delta function: ( δ ( = ( g t t t dt g t 0 0 Replication property of the delta function: the convolution of any function with the delta function leaves that function unchanged. ( δ ( ( τ δ ( τ τ ( g t t = g t d = g t 61

62 Chapter 2.5 Dirac Delta Function Fourier transform of the delta function is given by ( ( ( δ ( 1 F δ t = δ t exp j2π ft dt = 1 This relation states that the spectrum of the delta function δ(t extends uniformly over the entire frequency interval. We may view the delta function as the limiting form of a pulse of unit area as the duration of the pulse approaches zero. t 62

63 Chapter 2.5 Dirac Delta Function Applications of the Delta Function DC Signal If g ( t G ( f then G ( t g ( f 1 δ ( f δ ( t 1 By applying the duality property to the Fourier-transform pair of and noting that the delta function is an even function, we obtain ( 1 δ f DC signal is transformed in the frequency domain into a delta function. Another definition for the delta function: Delta functin is real. exp( j2π ft dt = δ( f cos( 2π ft dt = δ( f 63

64 Chapter 2.5 Dirac Delta Function Applications of the Delta Function Complex Exponential Function 1 δ ( f Sinusoidal Functions Applying the frequency-shifting property: exp( j 2π fct g( t G ( f f c ( j π f t δ ( f f exp 2 c c By using Euler sformula: 1 cos( 2π ft c = exp( j2π ft c + exp( j2π ft c cos( 2π fct δ( f fc + δ( f + fc sin 2π f t δ f f δ f + f 2 2 j ( ( ( c c c 64

65 Chapter 2.5 Dirac Delta Function Applications of the Delta Function Signum Function:sgn(t : Definition: + 1, t > 0 ( sgn t = 0, t = 0 1, t < 0 The signum function does not satisfy the Dirichlet conditions and does not have a Fourier transform. The signum function can be viewed as the limiting iti form of the antisymmetric double-exponential pulse as the parameter a approaches 0. ( at exp, t > 0 g( t = 0, t = 0 exp ( at, t < 0 65

66 Chapter 2.5 Dirac Delta Function Applications of the Delta Function Signum Function (cont. t From Example 2.3 for double exponential pulse G( f = a 2 j4π f + ( 2π f 2 F ( sgn ( t 4jπ f 1 1 = lim = sgn( t a 0 2 a + 2 f j π f j π f ( 2 π f 2 66

67 Chapter 2.5 Dirac Delta Function Applications of the Delta Function Unit Step Function ( u t 1, t > 0 1 =, t = 0 2 0, t < 0 1 = sgn ( ( t u t By using the linearity property of the Fourier transform and 1 1 j2π f + 2 ( δ ( f u t 1 δ ( f 67

68 Chapter 2.5 Dirac Delta Function Applications of the Delta Function Integration ti in the Time Domain (Revisited it Let t ( ( τ y t = g dτ The integrated signal y(t can be viewed as the convolution of the original signal g(t and the unit step function u(t, as shown by ( = ( τ ( τ τ = ( ( y t g u t d g t u t ( The time-shifted unit step function u t τ is defined d by 1, τ < t 1 u( t τ =, τ = t 2 0, τ > t 68

69 Chapter 2.5 Dirac Delta Function Applications of The Delta Function (cont. Integration ti in the Time Domain (Revisited it (cont. The Fourier transform of y(t can be easily obtained: Since Y f G f 1 1 j2π f 2 f ( = ( + δ ( ( δ ( = ( 0 δ ( G f f G f t Y f 1 1 j2π f G f 2 G f ( = ( + ( 0 δ ( 1 1 g d G f G f j2π f 2 ( τ τ ( + ( 0 δ( Eq. (2.39 is a special case of the above equation with G(0=0. 69

70 Chapter 2.6 Fourier Transform of Periodic Signals Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University

71 Chapter 2.6 Fourier Transform of Periodic Signals A periodic signal can be represented in terms of a Fourier transform provided that this transform is permitted to include delta functions. Consider a periodic signal g (tofperiodt T T 0 0: n= ( exp ( 2π g t = T c n j nf 0t 0 0 where c n is the complex Fourier coefficient defined by 1 T T T 2 ( exp( 2π 0 cn = g T T t j nf 0t dt and f 0 is the fundamental frequency f 0 =1/T 0. 71

72 Chapter 2.6 Fourier Transform of Periodic Signals Let g(t be a pulse like function, which equals g T (t over one 0 period and is zero elsewhere; that is, T0 T0 gt ( t, t 0 g( t = 2 2 0, elsewhere g (t may now be expressed T in terms of the function g(t 0 g T 0 ( ( g t = g t mt T 0 0 m= = g(t is Fourier transformable and can be viewed as a generating function, which generates the periodic signal g (t. T 0 1 T 2 ( exp( 2π ( exp( 2π ( 0 c = n g T T t j nf t dt = f g t j nf t dt f G nf 0 T where G(nf 0 is the Fourier transform of g(t at the frequency nf

73 Chapter 2.6 Fourier Transform of Periodic Signals The formula for the reconstruction of the periodic signal g T (t can 0 be rewritten as: c = f G ( nf n 0 0 ( exp( 2π ( exp( 2π g t = c j nf t = f G nf j nf t T0 n n= n= ( ( ( exp( 2π g t = g t mt = f G nf j nf t T m= n= The above equation is one form of Poisson s sum formula. ( j π f t δ ( f f exp 2 c c m= ( ( δ ( g t mt f G nf f nf ( n= The Fourier transform of a periodic signal consists of delta functions occurring at integer multiples of the fundamental frequency f 0 =1/T 0, including the origin, i and that each hdelta function is weighted by a factor equal to the corresponding value of G(nf 0. 73

74 Chapter 2.6 Fourier Transform of Periodic Signals The function g(t, constituting one period of the periodic signal g T (t, hasacontinuousa spectrum defined by G( f. 0 The periodic signal g T (t has a discrete spectrum. 0 Periodicity in the time domain has the effect of changing the frequency-domain description or spectrum of the signal into a discrete form defined at integer multiples of the fundamental frequency. 74

75 Chapter 2.6 Fourier Transform of Periodic Signals [Example 2.11] Ideal Sampling Function An ideal sampling function, or Dirac comb, consists of an infinite sequence of uniformly spaced delta functions. δ ( t δ ( t mt = T 0 0 m= The generating function g(t for the ideal sampling function δ (t T 0 consists of the delta function δ(t. We therefore have G( f =1 and G(nf 0 =1 for all n. Using Eq. (2.88 g ( t mt0 f0 G ( nf0 δ ( f nf0 yields m= δ m= n= ( t mt f δ ( f nf The Fourier transform of a periodic train of delta functions, spaced T 0 seconds apart, consists of another set of delta functions weighted by the factor f 0 =1/ T 0 and regularly spaced f 0 Hz apart along the frequency axis n= 75

76 Chapter 2.6 Fourier Transform of Periodic Signals [Example 2.11] Ideal Sampling Function (cont. From Poisson s sum formula: g ( t mt = f G( nf exp( j2π nf t m= n= δ( t mt0 = f0 exp( j2πnf0t Fourier exp( j2πmft0 = f0 δ( f nf0 m= n= Dual m= n=

77 Nyquist Sampling Theorem A band-limited signal of finite energy, which only has frequency components less than f m Hertz, is completely described by specifying the values of the signal at instants of time separated by 1/2 f m seconds. 1 TS or sampling rate f S 2 f m 2 f m A band-limited signal of finite energy, which only has frequency components less than f m Hertz, may be completely recovered from a knowledge of its samples taken at the rate of 2 f m samples per second. The sampling rate of 2f m per second, for a signal bandwidth of f m Hertz, is called the Nyquist rate; its reciprocal 1/2 f m (measured in seconds is called the Nyquist interval. 77

78 Nyquist Sampling Theorem 1 X S ( f δ = X ( f X ( f = X ( f nfs T S n= 78

79 Spectra for Various Sampling Rates 79

80 Chapter 2.7 Transmission of Signals Through Linear Systems Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University

81 Chapter 2.7 Transmission of Signals Through Linear Systems System: any physical device that produces an output signal in response to an input signal. Excitation: input signal. Response: output signal. In a linear system, the principle of superposition holds, i.e., the response of a linear system to a number of excitations applied simultaneously is equal to the sum of the responses of the system when each excitation is applied individually. Important examples: filters, communication channels. Filter: a frequency-selective device that is used to limit the spectrum of a signal to some band of frequencies. Channel: transmission medium that connects the transmitter and receiver of a communication system. 81

82 Chapter 2.7 Transmission of Signals Through Linear Systems Time Response In the time domain, a linear system is described din terms of fits impulse response, which is defined as the response of the system (with zero initial conditions to a unit impulse or delta function δ(t applied to the input of the system. If the system is time invariant, iant,then the eshape apeof the impulse puse response is the same no matter when the unit impulse is applied to the system. Convolution Integral: ( ( τ ( τ τ ( ( y t = x h t d = x t h t ( ( ( ( = h τ xt τ dτ = h t x t 82

83 Chapter 2.7 Transmission of Signals Through Linear Systems Causality and Stability Causal: A system is said to be causal if it does not respond before the excitation is applied. For a linear time-invariant i i (LTI system to be causal, the impulse response h(t must vanish for negative time, i.e. h(t=0, t<0. A system operating in real time to be physically y realizable, it must be causal. The system can be noncausal and yet physically realizable. Stable: A system is said to be stable if the output signal is bounded for all bounded input signals. Bounded input-bounded output (BIBO stability criterion. For a LTI system to be stable, the impulse response must be absolutely integrable, i.e. ( htdt 83 < (2.100

84 Chapter 2.7 Transmission of Signals Through Linear Systems Frequency Response Consider a LTI system of impulse response h(tdrivenbya a complex exponential input of unit amplitude and frequency f ( = exp ( 2 π xt j ft The response of the system is obtained as ( = ( ( = ( exp 2 ( yt = ht xt = h τ j π f t τ d τ ( ( ( = exp j2π ft h τ exp j2π fτ dτ Transfer function of the system is defined as the Fourier transform of its impulse response ( ( exp( 2π y( t = H( f exp( j2π ft y ( t = H ( f x ( t ( f H f h t j ft dt H ( exp( 2 ( x t = j π ft xt 84

85 Chapter 2.7 Transmission of Signals Through Linear Systems Frequency Response (cont. Consider an arbitrary signal x(t applied to the system: x( t = X ( f exp( j2π ft df or, equivalently, in the limiting form (a superposition of complex exponentials of incremental amplitude Because the h system is i linear, the h response is: i ( lim ( exp( 2π x t = X f j ft Δf Δf 0 f = kδf k = ( = lim ( ( exp( 2π y( t = H( f x( t y t H f X f j ft df Δ f 0 f = kδf k = ( ( exp( 2π ( = ( ( = H f X f j ft df ( ( exp( 2π Y f H f X f 85 y t = Y f j ft df

86 Chapter 2.7 Transmission of Signals Through Linear Systems Frequency Response (cont. The transfer function H( f is a characteristic property of a LTI system. It is a complex quantity: H( f H ( f exp jβ ( f = H( f : amplitude response β( f : phase or phase response If the impulse response h(t is real-valued, the transfer function H( f exhibits conjugate symmetry: H ( f = H ( f β ( f = β ( f 86

87 Chapter 2.7 Transmission of Signals Through Linear Systems Frequency Response (cont. Illustrating the definition of system bandwidth Low-pass system of bandwidth B Band-pass system of bandwidth 2B 87

88 Chapter 2.7 Transmission of Signals Through Linear Systems Paley-Wiener Criterion A necessary and sufficient condition for a function α( f to be the gain of a causal filter is the convergence of the integral α ( f df < 2 1+ f this condition is known as the Paley-Wiener criterion. We may associate with this gain a suitable phase β( f, such that the resulting filter has a causal impulse response that is zero for negative time. The Paley-Wiener criterion is the frequency-domain equivalent of the causality requirement. A realizable gain characteristic may have infinite attenuation for a discrete set of frequencies, but it cannot have infinite attenuation over a band of frequencies. 88

89 Chapter 2.8 Filters Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University

90 2.8 Filters A filter is a frequency-selective device that is used to limit the spectrum of a signal to some specified band of frequencies. Frequency response is characterized by a passband and a stopband. The frequencies inside the passband are transmitted with little or no distortion, whereas those in the stopband are rejected. There are low-pass, high-pass, band-pass, and band-stop filters. low-pass H ( f H ( f band-pass high-pass f 0 0 H ( f band-stop H ( f f f f

91 2.8 Filters Time response of the ideal low-pass filter xt ( ( = ( y t x t t 0 The transfer function of an ideal low-pass filter is defined dby: H ( f ( exp j2π ft0, B f B = 0, f > B The ideal low-pass filter is noncausal because it violates the Pl Paley-Wiener criterion. i This can be confirmed by examining the impulse response h(t B ( = exp 2π ( 0 B ht j f t t df Bt ( t0 ( t t sin 2π = = 2 Bsinc 2 B t t π 0 ( 0 (

92 2.8 Filters There is some response from the filter before the time t=0, so confirming that the ideal low-pass filter is noncausal. However, we can make the delay t 0 large enough such hthatt ( sinc 2 B t t0 1 for t < 0 By so doing, we are able to build a causal filter that closely l approximates an ideal low-pass filter. 92

93 2.8 Filters [Example 2.13] Pulse response of ideal low-pass filter Consider a rectangular pulse x(t ( of unit amplitude and duration T, which is applied to an ideal low-pass filter of bandwidth B.The problem is to determine the response y(t of the filter. Using Eq. (2.118, and setting t 0 =0 for simplification ht ( = 2 B sinc ( 2 Bt Gibbs phenomenon the resulting filter response ( = ( ( y t x τ h t τ dτ = 2B T T 2 sin 2π B t 2 2πB t (no closed form ( τ ( τ dτ 93

94 2.8 Filters Design of Filters Design of filters is usually carried out in the frequency domain. There are two basic steps: The approximation of a prescribed frequency response(i.e. amplitude response, phase response, or both by a realizable transfer function. The realization of the approximating transfer function by a physical device. For an approximating transfer function H( f to be physically realizable, it must represent a stable system. Stability is defined here on the basis of the bounded input bounded output criterion described in Eq. ( In the following, we specify the corresponding condition for stability in terms of the transfer function. The traditional approach is to replace j2π f with s. 94

95 2.8 Filters Design of Filters Ordinarily, the approximating transfer function H (s isa rational function, which may be expressed in a factored form as: ' H s = H f ( ( j2πf = s = K ( s z 1 ( s z 2 ( s zm ( s p ( s p ( s p 1 2 n where K is scaling factor; z 1, z 2,, z m are the zeros of the transfer function; p 1, p 2,,,pp n are its poles. For low-pass and band-pass filters: m<n. If the system is causal, all the poles of the transfer function H (s should be inside the left half of the s-plane, i.e. Re[p i ]<0. 95

96 2.8 Filters Different Types of Filters Two popular families of low-pass filters: Butterworth filters and Chebyshev filters. All their zero are at s= and the poles are confined to the left half of the s-planes plane. Butterworth filter The poles of the transfer function lie on a circle with origin as the center and 2πB as the radius, where B is the 3-dB bandwidth of the filter. Is said to have a maximally flat passband response. Chebyshev filter The poles lie on an ellipse. Provide faster roll-off than Butterworth th filter by allowing ripple in the frequency response. Type 1 filters have ripple only in the passband. Type 2 filters have ripple only in the stopband and are seldom used. 96

97 2.8 Filters Comparison of the amplitude response of 6 th order Butterworth low-pass filter with that of 6 th order Chebyshev filter. 97

98 2.8 Filters A common alternative to both the Butterworth and Chebyshev filters is the elliptic filter, which has ripple in both the passband and the stopband. Elliptic filter provide even faster roll-off for a given number of poles but at the expense of ripple in both the passband and stopband. Butterworth filters are the simplest and elliptic filters are the more complicated to design in mathematical terms. The finite-duration impulse response (FIR filter is often used in digital signal processing. The FIR filter is the equivalent of the tapped delay-line filter described in the previous section. The FIR filter has only zeros; it is thus inherently stable. 98

99 2.8 Filters Amplitude response of 8 th order elliptic bandpass filter. 99

100 2.8 Filters Amplitude response of 29-tap FIR low-pass filter. 100

101 Chapter 2.9 Low-Pass and Band-Pass Signals Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University

102 2.9 Low-Pass and Band-Pass Signals Communication using low-pass signals is referred to as baseband communication. In some transmission media, there is insufficient spectrum at baseband (e.g., radio waves or the properties p of media are not conductive to conducting signal at baseband (e.g., optical fibers. In these cases, we employ band-pass communications. Illustration of spectrum of band-pass signal. Illustration of time-domain band-pass signal. 102

103 2.9 Low-Pass and Band-Pass Signals Narrow-band signal: the bandwidth 2W is small compared to the carrier frequency f 0. A real-valued band-pass signal g(t with non-zero spectrum G( f in the vicinity of f c may be expressed in the form: ( = ( + φ ( g t a t cos 2πf c t t a(t: envelope (non-negative φ( t : phase Using the relationship cos(a+b=cos(acos(b-sin(asin(b I ( = ( cos ( 2π ( sin ( 2π g t = g t ft g t ft (2.123 I c Q c ( = ( cos φ ( and ( = ( sinφ ( g t a t t g t a t t in-phase of g( t quadrature of g( t Q ( = ( + ( a t g t g t φ ( t I Q ( g 1 Q t = tan gi ( t

104 2.9 Low-Pass and Band-Pass Signals Complex Baseband Representation Eq. (2.123 may be written as g( t = Re g ( t exp( j2πf c t (2.126 where we define gt ( = gi ( t + jgq ( t The gi ( t and gq ( t are real, we refer to gt ( as the complex envelope of the band-pass signal. 1 g( t = g( t exp( j2πfct + g ( t exp( j2πfct 2 g ( t G ( f F 1 * G( f = G ( f fc + G ( f fc 2 1 ( G f G( f Re * ( A = ( A+ A 2 W 0 W f 104 f c 2W 0 f c f 2W

105 2.9 Low-Pass and Band-Pass Signals [Example 2.14] PF pulse (continued read by yourself To determine the complex envelope of fthe RF pulse t g t = A cos 2π f c t T Assume f c T>>1, so that g(t is narrow-band ( rect ( t g t = Re A rect exp j2πf c t T the complex envelope is t g ( t = A rect T and the envelope equals ( ( t a ( t = g ( t = A rect T 105

106 Chapter 2.10 Band-Pass Systems Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University

107 2.10 Band-Pass Systems Summaries of low-pass systems: x(t represents the message signal, y(t is the received or output signal, and h(t is the impulse response of the channel or filter. X( f =F[x(t], H( f = F[h(t], Y( f = F[y(t]. yt = xτ ht τ dτ Time domain ( ( ( Y( f = H( f X ( f Frequency domain These equations are valid for linear systems. Band-pass system yt = xτ ht τ dτ Time domain ( ( ( Frequency domain Y ( f = H ( f X ( f 107

108 2.10 Band-Pass Systems Band-pass systems ( ( ( g t = Re g t exp j2π fct (2.126 When h(t is the impulse response of a bandpass filter, by analogy with g(t of Eq , it may be represented as ht ( = Re ht ( exp( j2πft c where h ( t is the complex impulse response of the bandpass filter. This response and its Fourier transform may be expressed as h( t = Re h ( t exp( j2πf ct 1 Re 2 1 * = h t exp j 2π f ct + h t exp j 2π fc t 2 1 * H ( f = H + H ( c ( c 2 f f f f * ( A = ( A+ A ( ( ( ( Positive frequency portions of H( f Negative frequency portions of H( f 108

109 2.10 Band-Pass Systems Since H ( f is low-pass limited to f <B, we can obtain ( 2 H f, f 0 H > ( f fc = 0, f < 0 0, f > 0 H ( f f c = * 2 H ( f, f < 0 This low-pass filter response is the frequency-domain equivalent of the complex impulse response of the filter. The output y(t is also a band-pass signal: ( ( ( ( π yt = Re ytexp j2 ft c where y t is the complex envelope of y(t. 109

110 2.10 Band-Pass Systems Y( f = H( f X ( f 1 * 1 * = H ( f fc + H ( f fc X ( f fc + X ( f fc * * = H ( f fc X ( f fc + H ( f fc X ( f fc * * = Y ( f fc + Y ( f fc H ( f fc X ( f fc 2 * = H X where ( 1 f fc f fc Y f = H ( f X ( f ( = 0 y ( t = h ( t x ( t ( The complex envelope of the band-pass output is the convolution of the complex envelope of the filter and the input, scaled by the factor ½. 110 ( (

111 2.10 Band-Pass Systems The analysis of a band-pass system is complicated due to the multiplying py factors cos(2πf f c t and sin(2πf f c t. The significance of Eq. (2.140 is that, we need only concern the low-pass functions, x ( t, h ( t, and y ( t. In other words, the analysis of a band-pass system is replaced by an equivalent but much simpler low-pass analysis that completely retains the essence of the filtering process y t h t x t 2 ( = ( ( 2 111

112 2.10 Band-Pass Systems [Example 2.15] Response of an ideal band-pass filter to a pulsed RF wave Target: compute the response of an ideal band-pass filter H( f to an RF pulse of duration T and carrier frequency f c ( f c T>>1 t x( t = A rect cos( 2πf c t T low-pass equivalent H 2, B< f < B = 0, f > B F ( f h ( t = 4 B sinc ( 2Bt 112

113 2.10 Band-Pass Systems low-pass equivalent ( x t t = A rect cos 2πf c t T x t t = A rect T ( ( 1 y t = h t x t 2 ( ( ( (no closed form 113

114 Chapter 2.11 Phase and Group Delay Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University

115 2.11 Phase and Group Delay Whenever a signal is transmitted through a dispersive (frequency- selective device such as a filter or communication channel, some delay is introduced into the output signal in relation to the input signal. In an ideal filter, the phase response varies linearly with frequency inside the passband of the filter, in which case the filter introduces a constant delay. Question: what if the phase response of the filter is nonlinear? Signal Models: assume that a steady sinusoidal signal at frequency f c is transmitted through a dispersive channel that has a total phase-shift of β( f c. Phase delay of the channel: β( f c /2 πf c [sec] is the time taken by the received signal phasor to sweep out this phase lag. Phase delay is not necessarily the true signal delay. The true signal delay is represented by the envelope or group delay. 115

116 2.11 Phase and Group Delay Assume that the dispersive channel is described by the transfer function: H( f = K exp jβ ( f where K is a constant the phase β( f is a nonlinear function of frequency. The input signal x(t consists of a narrow-band signal: ( ( ( x t = m t cos 2πf c t where m(tisalowpass a low-pass (information-bearing signal with its spectrum limited to the frequency interval f W. Assume f c >> W. By using the Taylor series about the point f=f f c and retaining only the first two terms: β ( f β( f + ( f f β( ( f β f c c f = f c 116 f Taylor series at n= 0 β f = ( n ( fc ( f f n! c n f c

117 2.11 Phase and Group Delay ( f β c Define phase delay: τ p = 2 π fc 1 β ( f Define group delay: τ = g c f = 2π f f β ( f β ( f β( fc + ( f fc f = f β( f 2π f 2 ( c cτ p π f fc τg ff The transfer function of the channel takes the form: ( ( H f Kexp j 2π f 2 cτ p j π f fc τ g Equivalent low-pass filter H f fc ( 2 exp( 2π τ 2π τ H f K j f j f c p g ( Low-pass complex envelope and its Fourier transform: x ( t = m( t, X ( f = M ( f F m( t 117 ( 2 H f, f > 0 = 0, f < 0

118 2.11 Phase and Group Delay The Fourier transform of the complex envelope of the received signal: 1 Y ( f = H ( f X ( f ( ( π τ ( π τ ( K exp j2 f exp j2 f M f c p g The term exp(-j2πfτ j f g M( f represents the Fourier transform of the delayed signal m(t-τ g. Complex envelope of the received signal: ( ( π τ ( τ y t Kexp j2 fc p m t g Received signal: yt ( = Re yt ( exp( j2π ft c ( τ cos 2π ( τ = Km t f t g c p 118

119 2.11 Phase and Group Delay The sinusoidal carrier wave cos(2πf c t is delay by τ p seconds, hence τ p represents the phase delay. Sometimes, τ p is also referred to as the carrier delay. The envelope m(t ( is delayed by τ g seconds; hence, τ g represents the envelope or group delay. τ g is related to the slope of the phase β( f, measured at f=f c. When the phase response β( f varies linearly with frequency, the signal is delayed but undistorted. When this linear condition is violated, we get group delay distortion. 119

120 Chapter 2.12 Sources of Information Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University

121 2.12 Sources of Information An example of waveform that represents an analog source of information. 121

122 2.12 Sources of Information Some source are digital in the sense that the information can be naturally represented as a sequence of zeros and ones. The digital waveform can be represented as: K = k k = 0 ( ( g t b p t kt Figure 2.38 (a rectangular pulse shape Figure 2.38 (b non-rectangular pulse shape 122