Representation of Signals & Systems

Transcription

1 Representation of Signals & Systems Reference: Chapter 2,Communication Systems, Simon Haykin. Hilbert Transform Fourier transform frequency content of a signal (frequency selectivity designing frequency-selective filters for the separation of signals on the basis of frequency contents) Here we use phase selectivity uses phase shifts between signals under consideration to achieve separation. Simplest phase shift is 80 0 polarity reversal in the case of a sinusoidal signal. Requires the use of an ideal transformer in general. Another shift of interest is 90 0 the resulting function of time is known as the Hilbert transform of the signal. Consider a signal g( with Fourier transform G(f). The Hilbert transform of g( is defined by gˆ ( g( ) d = g( t t, - convolution () This is a linear operation. The inverse Hilbert transform is defined by gˆ( ) g( d = gˆ ( - t The functions g( and gˆ ( are said to constitute a Hilberttransform pair. t (2)

2 In terms of Fourier transform; -j sign(f) t Thus Gˆ ( f ) jsgn( f ) G( f ) So for a given signal g( we may obtain its Hilbert transform gˆ ( through a two port device with transfer function -j sign(f). It may be considered as giving phase shift for all positive frequencies and for all negative frequencies. The amplitudes of all frequency components in the signal are unaffected. Such a device is called as a Hilbert transformer Phase of H(f) f Hilbert transform is used to: realize phase selectivity in SSB modulation. provide the mathematical basis for the representation of band-pass signals. The Hilbert transform as defined above applies to any signal that is Fourier transformable. Thus it may be applied to energy signals as well as power signals. e.g. Sinusoidal functions 2

3 Consider g(= cos(2f c G(f)=/2. [(f-f c ) + (f+f c )] Therefore G ˆ( f ) = -j sgn(f).g(f) = -j/2. [(f-f c ) + (f+f c )]sgn(f) = /2j. [(f-f c ) - (f+f c )] This is the Fourier transform of sin(2f c. So the H.T. of the cosine function is sin(2f c. Similarly sin(2f c has a H.T. equal to - cos(2f c. Properties of the Hilbert Transform Hilbert transform differs from the Fourier transform as it operates only in time domain. The signal is usually assumed to be real valued.. A signal g( and its Hilbert transform have the same amplitude spectrum. -j sgn(f) = f. 2. If gˆ ( is the H.T. of g(, then the H.T. of gˆ ( is g(. (check!!) 3. A signal g( and its H.T. are orthogonal. Here we use multiplication theorem in F.T. g ( gˆ( dt G( f ) Gˆ( f ) df =j G ( f ) Gˆ ( f ) df 3

4 =j sgn( =j sgn( =j sgn( f ) G( f ) G( f ) df f ) G( f ) G * ( f ) df f ) G( f ) where G(-f) = G*(f) for g( real. The integrand is odd. Thus g ( gˆ( dt 0. Similarly we may show that a power signal g( and its Hilbert transform gˆ ( are orthogonal over one period; 2 df lim T 2T T T g( gˆ( dt 0. (check the earlier example) 4

5 2. Pre-Envelope Consider a real-valued signal g(. We define the pre-envelope of the signal g( as the complex-valued function g + (= g( + j g ˆ( where gˆ ( is the H.T. of g(. We note that the given signal g( is the real part of the preenvelope g + ( and the H.T. of the signal is the imaginary part of the pre-envelope. Pre-envelope makes the handling of band-pass signals and systems easier. The Fourier transform: Thus G + (f) = G(f) + j [-jsgn(f)]g(f) 2G( f ), f 0 G + (f) = G(0), f 0 0, f 0 Where G(0) is the value of G(f) at frequency f=0. Therefore 0 g ( 2 G( f )exp( j2 f df G(f) G(0) -W W f 5

6 G + (f) 2G(0) W f In the above case we used a low-pass signal but pre-envelope can be defined for any signal with a spectrum. We may define the pre-envelope for negative frequencies as g - (= g( - j g ˆ( The two pre-envelopes g + ( and g - ( are complex conjugates of each other. G - (f) = 0, f 0 G(0), f 0 2G( f ), f 0 6

7 3. Canonical Representation of Band-Pass Signals We say that a signal g( is a band-pass signal if its Fourier transform G(f) is non-negligible only in a band of frequencies of total extent 2W, e.g, centered about some frequency f c. We refer to f c as carrier frequency. In most cases 2W is small compared with f c narrow-band signal. Let the pre-envelope of such a signal be expressed as g + (= g ( t ) exp(j2fc We refer to g ( t ) as the complex envelope of the signal. The spectrum of g + ( is limited to the band f c -W f f c +W. We find that the spectrum of g ( t ) is therefore limited to the band -W f W and centered at the origin. The complex envelope g ( t ) of a band-pass signal g( is a low-pass signal. G(f) f -f c 2W G ( f ) 2 G(f c ) f c 2W -W W f 7

8 g( = Re[g + (]=Re [ g ( t ) exp(j2fc ] In general, g ( t ) is complex. To emphasize this we usually express it as g ( t ) = gi ( + j g Q ( where g I ( and g Q ( are both real-valued low-pass functions. The low-pass property is inherited from the complex envelope g ( t ). Therefore we can express the original band-pass signal in the canonical form as g( = g I (cos(2f c - g Q ( sin(2f c We refer to g I ( as the in-phase component of the band-pass signal g( and to g Q ( as the quadrature component of the signal. We can now give the corresponding phasor diagrams. Also the following schemes can be given to obtain g I ( and g Q ( from g( and vice-versa. The multiplication of low pass in phase component g I ( by cos(2f c and the multiplication of low pass quadrature component g Q ( by sin(2f c represent linear forms of modulation. Given that the carrier frequency f c is sufficiently large, g( is referred to as a pass-band signaling waveform. Correspondingly the mapping from g I ( and g Q ( into g( is known as passband modulation. 8

9 We can express g ( t ) in polar form as g ( t ) = a( exp [j(] where a( and ( are both real-valued low-pass functions. Based on this polar representation, the original band-pass signal g( is defined by g( = a(cos [2f c t + (] We refer to a( as the natural envelope or just envelope of bandpass signal g( and to ( as the phase of the signal. The above equation represents a hybrid form of amplitude modulation and angle modulation. The information content of the signal g( is completely represented by the complex envelope g ( t ). 9

10 4 Band-Pass Systems Already know how to represent band-pass signals. analyze band-pass systems based on the relation between low-pass and band-pass systems. (Hilbert transform) Consider a narrow-band signal x( with Fourier transform X(f). Assume that the X(f) is limited to W Hz of the carrier frequency f c. Also assume that W < f c. Let x( = x I (cos(2f c - x Q ( sin(2f c Where x I ( in-phase component, x Q ( quadrature component. The complex envelope of x(, x ( t ) = xi ( + jx Q ( Let x( be applied to a LTI (linear time-invarian band-pass system with impulse response h( and transfer function H(f).H(f) is limited to B Hz of the carrier frequency f c. The system bandwidth 2B is either narrower or equal to the i/p signal bandwidth 2W. We want to express h( in terms of h I ( and h Q ( its in-phase and quadrature components. Thus, h( = h I (cos(2f c - h Q ( sin(2f c Define the complex impulse response of the band-pass system as h ( t ) = hi ( + jh Q ( 0

11 So h( = Re[ h ( t ) exp(j2fc ] Note that h I (, h Q (and h ( t ) are all low-pass functions limited to B f B. 2h( = h ( t ) exp(j2fc + h * ( exp(-j2f c h * ( is the complex conjugate of h ( t ). Taking Fourier transform, 2H(f) = H( f f ) H *( f ) c f c Now for a real impulse response h(, H*(f) = H(-f). As H ( f ) represents a low pass function limited to f B with B < f c we can obtain from the above relation: H( f ) = 2H(f) f > 0 f c So we can find H ( f ) by taking the positive frequency part of H(f) and shifting it to origin scaled by 2. Then taking the inverse Fourier transform of the complex impulse response h ( t ). H ( f ) we can find Without loss of generality we assume that X(f) and H(f) are both centered around f c. Let y( be the output signal. It is also a bandpass signal. Hence, y(= Re [ y ( t ) exp(j2fc ] Also y( = h ( ) x( t ) d

12 In terms of pre-envelopes, we have h(= Re [h + (] x( = Re [x + (] Therefore y( = Re[ h )] ( )]Re[ x ( t d It can be shown that * Re[ h ( )]Re[ x ( )] d Re h ( ) x ( ) d 2 As we use x(-) instead of x() we can avoid the conjugate. Thus y(= Re 2 h ( ) x ( t ) d = Re 2 h ( )exp( j2f c ) x( t )exp( j2f c ( t )) d = Therefore, Re 2 exp( j2f c h ( ) x( t ) d y t h ) 2 ( ) ( x( t ) d = h( x ( = [ h I ( + j h Q (] [ x I ( + j x Q ( ] 2

13 If y ( t ) = yi ( + j y Q ( Then 2y I ( = h I ( x I ( h Q ( x Q ( 2y Q ( = h Q ( x I ( + h I ( x Q ( Therefore, for the purpose of obtaining the in-phase and quadrature components of the complex envelope y ( t ) of the system output, we can use the low-pass equivalent model shown below. x I ( h I ( h Q ( + - 2y I ( x Q ( h Q ( h I ( + + 2y Q ( 3