Representation of Signals and Systems

Lecturer: David Shiung

1 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 Abstract (2/2)

Complex representation of signals and systems

Rotation of signal

In-phase and quadrature components of a band- pass signal

3 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 Fourier- transform pair

4 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 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 Properties of the Fourier Transform (1/2)

Table A6.2

7 Properties of the Fourier Transform (2/2)

8 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 Fourier Transforms of Periodic Signals (1/2)

A periodic signal gT0(t) of period T0 ; we represent gT0(t) in terms of the complex exponential Fourier series:

f0 is the fundamental frequency with

Let

10 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 Fourier-transform Pairs (1/2)

Table A6.3

12 sinc( 2Wt)=sin( 2πWt)/ 2πWt   ,1 f > 0  sgn( f ) =  ,0 f = 0  − ,1 f < 0  13 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 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 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 Frequency Response of Linear Time-invariant Systems (2/2)

For real-valued impulse response h(t) :

17 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 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.

A signal cannot be strictly limited in both time and frequency.

18 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 Bandwidth (3/3)

Root mean square (rms) bandwidth:

20 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 Noise Equivalent Bandwidth (1/1)

Illustrating the definition of noise-equivalent bandwidth for a low-pass filter

22 Hilbert Transform (1/2)

Phase characteristic of linear two-port device for obtaining the Hilbert transform of a real-valued signal

23 Hilbert Transform (2/2)

The Hilbert transform of g(t) is

The inverse Hilbert transform:

Fourier transform of 1/π t

signum function: 24 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 Complex Representation of Signals and Systems (1/2)

Illustrating an interpretation of the complex envelope 26 and its multiplication by exp( j2π fct) 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 Problems

Prove the Fourier-transform pairs on this slide (pp. 12-13).

Prove the three properties of the Hilbert transform on this slide (p.25).