ENSC327 Communications Systems 2: Fourier Representations

1

ENSC327

Communications Systems

2: Fourier Representations

Jie Liang

School of Engineering Science

Simon Fraser University

2

Outline

�Chap 2.1 – 2.5:

� Signal Classifications

� Fourier Transform

� Dirac Delta Function (Unit Impulse)

� Fourier Series

� Bandwidth

� (Chap 2.6-2.9 will be studied together with Chap. 8)

3

Signal Classifications:

Deterministic vs Random

� Deterministic signals: can be modeled as completely

specified functions, no uncertainty at all

� Example: x(t) = sin(a t)

x(t)

Random noise e(t)

y(t): random

� Random signals: take random value at any time

� Example: Noise-corrupted channel output

� Probability distribution is needed to analyze the signal

� It is more useful to look at the statistics of the signal:

� Average, variance …

4


Periodic vs Aperiodic

� Periodic: A signal x(t) is periodic if and only if we can find some constant T0 such that x(t+T0) = x(t), -∞< t <∞.

� Fundamental period: the smallest T0 satisfying the equation above.

� Aperiodic: Any signal not satisfying the equation is called aperiodic.

……

t t + T0

5


Energy Signals vs Power Signals

� Power and energy of arbitrary signal x(t):

∫∫∞

∞−−∞→

== dttxdttxET

TT

22

)()(lim

∫−

∞→

=

T

TT

dttxT

P2

)(2

1lim�Power:

�Energy:

Power is the average amount of energy transferred

per unit of time.

=PFor a periodic signal:

=EWhat’s its energy?

6

0 (so 0)E P< < ∞ =

� A signal is called Energy Signal if its energy is finite

0 (so E )P< < ∞ = ∞

� A signal is called Power Signal if its power is finite


Energy Signals vs Power Signals

� Periodic signals are power signals, but not energy signals.

� x(t) = 0 at infinity (will be used later)

7

Outline



�Delta Function

� Fourier Series

�Bandwidth

8

Types of Fourier Series and Transforms

� Continuous-time signals:� 1. Aperiodic:

� 2. Periodic:

� Discrete-time signals:

� 3. Aperiodic:

� 4. Periodic:

Continuous-

time signalsDiscrete-time

signals

Aperiodic Periodic

1 2

Aperiodic Periodic

3 4

9

Fourier Transform (FT)

� For aperiodic, continuous-time signal:

� In terms of frequency f: Recall ω = 2 π f

10

Amplitude and Phase Spectra

� Important Property: If g(t) is real, then G(f) is

conjugate symmetric:

� Proof: (* is the complex-conjugate operator)

).()( ,)()(or ),()(* fffGfGfGfG −−=−=−= θθ

spectrum phase :)(

spectrum amplitude :)(

f

fG

θ

)()()( fjefGfG θ=

11

Properties of FT

�Conjugation rule:

(The notation means that

are FT pairs)

� Proof:

)( )(* -fGtg *↔

)( and )(* -fGtg * ↔

12

Properties of FT

� Symmetry:

� If g(t) is real and even, then G(f) is real and even.

� If g(t) is real and odd, then G(f) is img. and odd.

� Proof (first statement only):

13

Example of Symmetry

�Unit rectangular function (or gate function):

[ ]

−∈

=otherwise. ,0

,0.5 ,5.0 ,1)(rect

t

t

-0.5 0.5

1

-4 -3 -2 -1 0 1 2 3 4-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

14

Example of Symmetry

�Rectangular Pulse:

)rect( T

tAg(t) =

|G(f)|

This is a special case of

the dilation property (next)

15

Properties of Fourier Transform

�Dilation:

� Proof:

� Compress (expand) in time � expand (compress) in frequency

)(1

)(a

fG

aatg ↔ (a is a real number)

16


� Another example: Given

)rect( T

tAg(t) =

� Applications of

)( ↔−tg

If in addition g(t) is real, G(f) is conjugate symmetric,

)( ↔−tg

)(sinc )(rect ft ↔

Find the FT of

)(1

)(a

fG

aatg ↔

17


�Duality:

� Proof:

)( )( fgtG −↔

18


�Duality:

�Example:

)( )( fgtG −↔

).( Find ).2(sinc )( fGWtAtg =

19

Uncertainty Principle of the FT

Narrow in time

Wide in frequency

Wide in time

Narrow in frequency

20


�Duality will be used later when we study

single sideband (SSB) communications and

Hilbert transform

<−

=

>

=

0. t,1

0, t,0

0, t,1

sgn(t)fjπ

1 ↔

Proof By Duality :

21


�Time shifting (delay):

Delayg(t) g(t – t0)

G(f)

02

0)( )(

πft-jefGttg ↔−

02πft-je

02

)(πft-j

efG

� Proof:

Time delay only affects the phase spectrum.

22


� Frequency Shifting:

�Very useful in communications

f

|X(f)|

0

f

|X(f - f 0)|

0

Low freq signal

)(G )(0

20 ffetgtfj

−↔π

23


�Example: )2cos()()( tfT

trecttg

cπ=

24


�Differentiation:

�This property is used in FM demodulation

( ) )(2 )(

fXfjdt

txd n

n

n

π↔

25


� Convolution: the convolution describes the input-output relationship of a linear time-invariant (LTI) system

� The convolution of two signals is defined as

� The formula is related to the properties of LTI system and impulse response.

� Note: it is very easy to make mistake about this formula. Pleasebe very careful, as it will appear in the exam.

� More on this in the end of this lecture.

=y(t)

26


� Convolution property: one of the most useful properties of FT

),()( ),()(Let 2211fGtgfGtg ↔↔

∫∞

∞−

↔− )()( )()(then 2121fGfGdtgg τττ

Proof:

Time domain convolution � frequency domain product

27


�Modulation: ),()( ),()(2211fGtgfGtg ↔↔

λλλ∫∞

∞−

−↔ dfGGtgtg )()()()(2121

Proof:

Time domain product � frequency domain convolution

28

Rayleigh’s Energy Theorem

(Parseval’s Theorem)

Can calculate the energy in either domain.

Proof:

∫∫∞

∞−

∞

∞−

= dffGdttg22

)()(

29

Outline



�Dirac Delta Function (Unit Impulse)

� Fourier Series

�Bandwidth

30

Dirac Delta Function (Unit Impulse)

� The Dirac delta function δ(t) is defined to satisfy two

relations:

1)(

.0for 0)(

=

≠=

∫∞

∞−

dtt

tt

δ

δ

�� δ(t) is an even function: δ(-t) = δ(t) .

� The definition implies the sifting property:

� Delta function can be defined by sifting property directly.

)(tδ

=−∫∞

∞−

dttttg )()(0

δ

)(tg

0t

)(0tg

)(0tt −δ

31

Dirac Delta Function (Unit Impulse)

� Since δ(t) is even function, we can rewrite this as

� Changing the variables, we get the convolution:

�� The convolution of δ(t) with any function is that

function itself.

� This is called the replication property of the delta

function.

32

Linear and time-invariant system

� Linearity: a system is linear if the input

leads to the output ,

where y1, y2 are the output of x1 and y2 respectively.

x(t) y(t)

)()(2211txatxa +

)()(2211tyatya +

� Time-invariant system: a system is time-invariant if

the delayed input has the output ,

where y(t) is the output of x(t).

A system is LTI if it’s both linear and time-invariant.

)(0ttx − )(

0tty −

33


� A linear and time-invariant system is fully

characterized by its output to the unit impulse, which

is called impulse response, denoted by h(t).

� The output to any input is the convolution of the input

with the impulse response:

)(tδ )(th

∫∞

∞−

−= τττ dthxty )()()(

34


� Proof of the convolution expression:

We start from the sifting property:

� This can be viewed as the linear combination of

delayed unit impulses.

� By the properties of LTI,

the output of x(t) will be the linear combination of

delayed impulse responses:

35

Fourier Transform of the delta

function

=∫∞

∞−

−

dtetftj π

δ2)( property, sifting By the

This is another example to demonstrate the Uncertainty Principle.

The FT of the delta func is

36

Applications of Delta Function

� FT of DC signal:

(i.e., DC signal only has 0 frequency component).

� Proof: Applying duality to 1G(f) )()( =↔= ttg δ

).()( 1g(t) ffG δ=↔=

37


� FT of :

� (Intuition: a pure complex exponential signal only has

one frequency component)

� Proof:

tfje

02π ).(

0

20 ffetfj

−↔ δπ

38


� FT of : tf02cos π

� FT of : tf02sin π

39

Outline



�Delta Function

� Fourier Series

�Bandwidth

40

Fourier Series Definition

� Suppose x(t) is periodic with period T0:

0 0 0( ) , o

jk t

k

k

x t X e t t t Tω

∞

=−∞

= ≤ < +∑1

( ) o

o

jk t

kT

o

X x t e dtT

ω−

= ∫

� Represent x(t) as the linear combination of fundamental signal

and harmonic signals (or basis functions)

� Xk: Fourier coefficients. Represent the similarity between x(t)

and the k-th harmonic signal.

:/2Let 00Tπω =

41

Fourier Series (cont.)

� Example: Find the Fourier series expansion of

� Method 1: use the definition

2

0 0( ) cos( ) sin (2 )x t t tω ω= +

1( ) o

o

jk t

kT

o

X x t e dtT

ω−

= ∫

( ) ojk t

k

k

x t X eω

∞

=−∞

= ∑

� Method 2: use trigonometric identity and Euler’s theorem:

42

Outline



�Delta Function

� Fourier Series

�Bandwidth

43

Definitions of Bandwidth (Chap 2.3)

� Bandwidth: A measure of the extent of significant

spectral content of the signal in positive frequencies.

� The definition is not rigorous, because the word “significant”

can have different meanings.

� For band-limited signal, the bandwidth is well-defined:

|X(f)|

f

W-W

Bandwidth is W.

|X(f)|

f

fc+Wfc-W

Bandwidth is 2W.

fc0

Low-pass Signals: Bandpass signals

44


�When the signal is not band-limited:

Different definitions exist.

� Def. 1: Null-to-null bandwidth

� Null: A frequency at which the spectrum is zero.

f

|X(f)|

0

Bandwidth is half of main lobe width

(recall: only pos freq is counted in bandwidth)

f

|X(f)|

0

Bandwidth = main lobe width

For low-pass signals: For Bandpass signals:

45


� Def. 2: 3dB bandwidth

f

|X(f)|

0

bandwidth

f

|X(f)|

0

bandwidth

A2/A

Low-pass Signals Bandpass signals

2

)( fX drops to 1/2 of the peak value, which corresponds to 3dB

difference in the log scale.

A2/A

dB35.0log1010

−=

46


� Def. 3: Root Mean-Square (RMS) bandwidth2/1

2

22

)(

)()(

−=

∫

∫∞

∞−

∞

∞−

dffG

dffGffW

c

rms

spectrum. squared Normalized :)(

)()(

2

2

∫∞

∞−

=

dffG

fGfG

1.)( since =∫∞

∞−

dffG

The RMS bandwidth is the standard deviation of the squared

spectrum.

freq.center :cf

47

Definitions of Bandwidth

�Radio spectrum is a scarce and expensive

resource:

� US license fee: ~ $77 billions / year

�Communications systems should provide the

desired quality of service with the

minimum bandwidth.

ENSC327 Communications Systems 2: Fourier Representations

Documents