Representation of Signals and Systems. Lecturer: David Shiung

Transcription

1 Representation of Signals and Systems Lecturer: David Shiung 1

2 Abstract (1/2) Fourier analysis Properties of the Fourier transform Dirac delta function Fourier transform of periodic signals Fourier-transform pairs Transmission of signals through linear systems Frequency response of linear time-invariant systems Bandwidth Time-bandwidth product Noise equivalent bandwidth Hilbert transform Properties of the Hilbert transform 2

3 Abstract (2/2) Complex representation of signals and systems Rotation of signal In-phase and quadrature components of a bandpass signal 3

4 Fourier Analysis (1/3) g(t) is a nonperiodic deterministic signal, the Fourier transform of g(t) is j=, f denotes frequency Inverse Fourier transform: g(t) and G(f) are said to constitute a Fouriertransform pair 4

5 Fourier Analysis (2/3) Dirichlet s conditions (sufficient for existence of a Fourier transform): g(t) is single-valued, with a finite number of maxima and minima in any finite time interval g(t) has a finite number of discontinuities in any finite time interval g(t) is absolutely integrable Physical realizability is a sufficient condition for the existence of a Fourier transform 5

6 Fourier Analysis (3/3) All energy signals are Fourier transformable The terms Fourier transform and spectrum are used interchangeably G(f) is the magnitude spectrum of g(t); arg{g(f)} is the phase spectrum 6

7 Properties of the Fourier Transform (1/2) Table A6.2 7

8 Properties of the Fourier Transform (2/2) 8

9 Dirac Delta Function (1/1) Definition of Dirac delta function (unit function): Shifting property of delta function: Convolution of any function with the delta function leaves that function unchanged (replication property) Delta function is a limiting form of a pulse of unit area with duration of the pulse approaches zero 9

10 Fourier Transforms of Periodic Signals (1/2) A periodic signal g T0 (t) of period T 0 ; we represent g T0 (t) in terms of the complex exponential Fourier series: f 0 is the fundamental frequency with Let 10

11 Fourier Transforms of Periodic Signals (2/2) This is called the Poisson s sum formula Periodicity in the time domain has the effect of changing the frequency-domain description or spectrum of the signal into a discrete form at integer multiples of the fundamental frequency 11

12 Fourier-transform Pairs (1/2) Table A6.3 sinc(2wt)=sin(2πwt)/2πwt 12

13 sgn( f ) = 1, 0, 1, f > 0 f = 0 f < 0 13

14 Transmission of Signals Through Linear Systems (1/2) A system refers to any physical device that produces an output signal in response to an input signal In a linear system, the principle of superposition holds. 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. A linear system is described in terms of its impulse response (response of the system to a delta function) When the response of the system is invariant in time, it is called a time-invariant system. 14

15 Transmission of Signals Through Linear Systems (2/2) Response of a system, y(t), in terms of the impulse response h(t) by Convolution is cumulative ( 可交換的 ) 15

16 Frequency Response of Linear Time-invariant Systems (1/2) Define the frequency response of the system as the Fourier transform of its impulse response The response of a linear time-invariant system to a complex exponential function of frequency f is the same complex exponential function multiplied by a constant coefficient H(f) The frequency response H(f) : magnitude response β(f): phase response 16

17 Frequency Response of Linear Time-invariant Systems (2/2) For real-valued impulse response h(t) : 17

18 Bandwidth (1/3) We may specify an arbitrary function of time or an arbitrary spectrum, but we cannot specify both of them together. If a signal is strictly limited in frequency, the timedomain 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. A signal cannot be strictly limited in both time and frequency. 18

19 Bandwidth (2/3) There is no universally accepted definition of bandwidth; nevertheless, there are some commonly used definitions for bandwidth. Null-to-null bandwidth: spectrum of a signal is bounded by well-defined nulls (i.e., frequencies at which the spectrum is zero). 3-dB bandwidth: the separation between zero frequency, where the magnitude spectrum drops to of its peak value 19

20 Bandwidth (3/3) Root mean square (rms) bandwidth: 20

21 Time-bandwidth Product (1/1) Time-bandwidth product or bandwidth-duration product: (duration x bandwidth) = constant Whatever definition we use for the bandwidth of a signal, the time-bandwidth product remains constant over certain classes of pulse signals. 21

22 Noise Equivalent Bandwidth (1/1) Illustrating the definition of noise-equivalent bandwidth for a low-pass filter 22

23 Hilbert Transform (1/2) Phase characteristic of linear two-port device for obtaining the Hilbert transform of a real-valued signal 23

24 Hilbert Transform (2/2) The Hilbert transform of g(t) is The inverse Hilbert transform: Fourier transform of 1/πt signum function: 24

25 Properties of the Hilbert Transform (1/1) A signal g(t) and its Hilbert transform have the same magnitude spectrum If is the Hilbert transform of g(t), then the Hilbert transform of is -g(t) A signal g(t) and its Hilbert transform are orthogonal over the entire time interval (-, ) 25

26 Complex Representation of Signals and Systems (1/2) Illustrating an interpretation of the complex envelope and its multiplication by exp(j2πf c t) 26

27 Complex Representation of Signals and Systems (2/2) (a) Scheme for deriving the in-phase and quadrature components of a band-pass signal (b) Scheme for reconstructing the band-pass signal from its in-phase and quadrature components 27

28 Problems Prove the Fourier-transform pairs on this slide (pp ). Prove the three properties of the Hilbert transform on this slide (p.25). 28