Chapter 1 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/wavelet/continuous time...

Chapter 1 The Continue-Time Fourier Transform

Instructor: Hongkai Xiong (熊红凯) Distinguished Professor (特聘教授)

http://min.sjtu.edu.cn

Department of Electronic Engineering Department of Computer Science and Engineering

Shanghai Jiao Tong University

1.0 Introduction

1.1 The Continuous-Time Fourier Transform

1.2 The Fourier Transform for Periodic Signals

1.3 Properties of the Continuous-Time Fourier Transform

1.4 The Convolution Property

1.5 The multiplication Property

1.0 Introduction

1.0 Introduction • Fourier Series Representation

▫ It decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). The discrete-time Fourier transform is a periodic

• Fourier Transform ▫ A representation of aperiodic signals as linear

combinations of complex exponentials

• If we know the response of an LTI system to some inputs, we actually know the response to many inputs

If then • If we can find sets of “basic” signals so that

▫ We can represent rich classes of signals as linear combinations of these building block signals.

▫ The response of LTI Systems to these basic signals are both simple and insightful.

𝑥𝑥𝑘𝑘[𝑛𝑛] → 𝑦𝑦𝑘𝑘[𝑛𝑛]

� 𝑎𝑎𝑘𝑘𝑥𝑥𝑘𝑘[𝑛𝑛]𝑘𝑘 → ∑ 𝑎𝑎𝑘𝑘𝑘𝑘 𝑦𝑦𝑘𝑘[𝑛𝑛�

• Real Exponential Signals: when 𝐶𝐶 and 𝛼𝛼 are real numbers, e.g. ▫ growing exponential, when ▫ decaying exponential, when ▫ constant

Continuous-Time Complex Exponentials Signals and Sinusoidal Signals

teCtx α⋅=)(Where 𝐶𝐶 and 𝛼𝛼 are complex numbers

tetx 2)( =0>α0<α0=α

• Periodic Complex Exponential and Sinusoidal Signals: when 𝐶𝐶is real, 𝛼𝛼 is purely imaginary, e.g. then the fundamental period 𝑇𝑇0 = 2𝜋𝜋/𝜔𝜔0 [s] , angular frequency 𝜔𝜔0[rad/s], and frequency 𝑓𝑓0 = 𝜔𝜔0

2𝜋𝜋= 1/𝑇𝑇0[Hz]

Unless noted otherwise, in this course, we always call ω0 frequency

▫ Important periodicity property : ▫ 1) the larger the magnitude of 𝜔𝜔0 , the higher the oscillation

in the signal ▫ 2) the signal 𝑥𝑥(𝑡𝑡) is periodic for any value of 𝜔𝜔0

tjte tj00 sincos0 ωωω +=

)(21cos 00

0tjtj eet ωωω −+= )(

21sin 00

0tjtj ee

jt ωωω −−=

Euler’s Relation

• A general representation, when 𝐶𝐶 and 𝛼𝛼 are complex numbers, denoted as , then

is the envelop of the waveform is the oscillation frequency Example of real part of 𝑥𝑥(𝑡𝑡)

0,CC ωαθ jre j += ＝

)()( 00)( θωωθ ++ ⋅⋅=⋅⋅= tjrttjrj eeceectx

rtec ⋅0ω

Damped sinusoids 0<r

• Real Exponential Signals: when 𝐶𝐶 and 𝛼𝛼 are real numbers ▫ e.g. growing function, when 𝛼𝛼 > 1

Discrete-Time Complex Exponentials Signals and Sinusoidal Signals

Where 𝐶𝐶 and 𝛼𝛼 are complex numbers

nCnx α⋅=][

nnx 2][ =

▫ decaying function, when 0 < 𝛼𝛼 < 1

▫ constant, when 𝛼𝛼 = 1

▫ alternates in set , when 𝛼𝛼 = −1

( )nnx 2/1][ −=

{ }CC,−

• Complex Exponential and Sinusoidal Signals: when 𝐶𝐶 is real, 𝛼𝛼 is a point on the unit circle, e.g.

or Its periodicity property? Similar to that of continuous-time signals?

• A general representation, when 𝐶𝐶 , 𝛼𝛼 are complex

numbers, denoted as ,then • is the envelop of the waveform

njenx 0][ ω= )sin(),cos(][ 00 φωφω ++= nAnAnx

0,CC ωθ α jj ree ＝=)n(nn 00][ θωωθ +⋅⋅=⋅⋅= jjnj ercerecnx

nrc ⋅

• Periodicity Property of Discrete-time Complex Exponentials ▫ a) recall the definition of the periodic discrete-time signal 𝑥𝑥 𝑛𝑛 = 𝑥𝑥 𝑛𝑛 + 𝑁𝑁 for all 𝑛𝑛

▫ b)if it is periodic, there exists a positive integer 𝑁𝑁, which satisfies 𝑒𝑒𝑗𝑗𝜔𝜔0𝑛𝑛 = 𝑒𝑒𝑗𝑗𝜔𝜔0(𝑛𝑛+𝑁𝑁) = 𝑒𝑒𝑗𝑗𝜔𝜔0𝑛𝑛𝑒𝑒𝑗𝑗𝜔𝜔0𝑁𝑁so, it requires 𝑒𝑒𝑗𝑗𝜔𝜔0𝑁𝑁 = 1, i.e. 𝜔𝜔0𝑁𝑁 = 2𝜋𝜋𝑚𝑚

▫ If there exists an integer satisfying that 2𝜋𝜋𝑚𝑚/𝜔𝜔0 is an integer, i.e. 2𝜋𝜋/𝜔𝜔0 is rational number , 𝑥𝑥 𝑛𝑛 is periodic with fundamental period of N = 2𝜋𝜋𝑚𝑚/𝜔𝜔0 , where 𝑁𝑁,𝑚𝑚 are integers without any factors in common.

otherwise, 𝑥𝑥 𝑛𝑛 is aperiodic. Different from that of continuous exponentials

njenx 0][ ω=

• Another difference from that of CT exponentials since for any integer the signal is fully defined within a frequency interval of length : , for any integer

Distinctive signals for different 𝜔𝜔0 within any 2π region, i.e. for any integer m

Without loss of generalization, for , the rate of oscillation in the signal increases with increases from 0 to 𝜋𝜋 Important for discrete-time filter design!

nmjnj ee )2( 00 πωω += m

π2 ( ) ( )( ]ππ 12,12 +− mm m

( ) ( )( ]ππ 12,12 +− mm

( ]ππω ,0 −∈nje 0ω

• Comparison of Periodic Properties of CT and DT Complex Exponentials/ Sinusoids

Distinct signals for distinct value of

Identical signals for values of separated by multiples of

Periodic for any choice of Periodic only of for some integers and

Fundamental angular frequency Fundamental angular frequency , if m and N do not have any factors in common

Fundamental period Fundamental period

tjetx 0)( ω=njenx 0][ ω=

0ω0ω

0ω Nm /20 πω =0>N m

2ωπm

• Unit Impulse Sequence

• Unit Step Sequence

Discrete-Time Unit Impulse and Unit Step Sequences

[ ]u n

• Relationship

• Sampling Property

• Signal representation by means of a series of delayed unit samples

]1[][][ −−= nununδ

∑−∞=

mmnu ][][ δ ∑

][][/k

knnu δ

—1st difference

—running sum

][]0[][][ nxnnx δδ ⋅=⋅

][][][][ 000 nnnxnnnx −⋅=−⋅ δδ

∑ −⋅=k

knkxnx ][][][ δ

• Unit Step Function

Continuous-Time Unit Step and Unit Impulse Functions

( )u t

t0 Notes: 𝑢𝑢(𝑡𝑡) is undefined at 𝑡𝑡 = 0

Can we find counterpart of the unit impulse function in CT domain as that in DT domain ?

Does it exist satisfying the following relationship

]1[][][ −−= nununδ

∑−∞=

mmnu ][][ δ ∑

][][/k

knnu δ

—1st difference

—running sum

dttdut )()( =δ

∫ ∞−=

tdtu ττδ )()(

—1st derivative

—running sum

• Unit Impulse Function ▫ Since 𝑢𝑢(𝑡𝑡) is undefined at 𝑡𝑡 = 0, formally it is not

differentiable, then define an approximation to the unit step 𝑢𝑢∆(𝑡𝑡) ,which rises from 0 to 1 in a very short interval ∆

▫ So ▫ And

tudt )()( ∆∆ =δ

)(lim)(0

tt ∆→∆= δδ

Notes: the amplitude of the signal at 𝑡𝑡 = 0 is infinite, but with unit integral from to , i.e. from to

)(tδ∞− ∞ −0 +0

• Unit Impulse Function ▫ Dirac Definition

▫ We also call such functions as singularity function or

generalized functions, for more information, please refer to mathematic references

)(tδ∞− ∞ −0 +0

=∫∞

∞−

Notes: the amplitude of the signal at 𝑡𝑡 = 0 is infinite, but with unit integral from to , i.e. from to

• Relationship

• Sampling Property

• Scaling Property

dttdut )()( =δ

∫ ∞−=

tdtu ττδ )()(

)()0()()( txttx δδ ⋅=⋅

)()()()( 000 tttxtttx −⋅=−⋅ δδ

( ) )()( tkdt

tkud δ=

Can we represent x(t) by using a series of unit

samples as that for DT signal?

—1st derivative

—running sum

Introduction

• By exploiting the properties of superposition and time invariance, if we know the response of an LTI system to some inputs, we actually know the response to many inputs

If then

𝑥𝑥𝑘𝑘[𝑛𝑛] → 𝑦𝑦𝑘𝑘[𝑛𝑛]

� 𝑎𝑎𝑘𝑘𝑥𝑥𝑘𝑘[𝑛𝑛]𝑘𝑘 → ∑ 𝑎𝑎𝑘𝑘𝑘𝑘 𝑦𝑦𝑘𝑘[𝑛𝑛�

Introduction

• If we can find sets of “basic” signals so that ▫ We can represent rich classes of signals as linear

combinations of these building block signals. ▫ The response of LTI Systems to these basic signals

are both simple and insightful. • If we represent input signal as a linear combination of

these basic signals, then the output is the combination of the responses of such basic signals.

• Candidate sets of basic signal ▫ Unit impulse function ▫ Complex exponential/sinusoid signals.

][/)( nt δδntj ze /ω

• For example: x[n]=…x[-3] δ[n+3]+ x[-2] δ[n+2]+ …+x[0] δ[n] +x[1] δ[n-1]+… • i.e.: x[n] can be represented as the weighted sum

The Representation of Discrete-Time Signals in terms of Impulses

𝑥𝑥[𝑛𝑛] = � 𝑥𝑥[𝑘𝑘]𝛿𝛿[𝑛𝑛 − 𝑘𝑘]+∞

𝑘𝑘=−∞

Weight Basic signal

Convolution-Sum Representation of LTI Systems

• 1.Assume

and so

Unit impulse response

Time invariant

𝑥𝑥[𝑛𝑛] = � 𝑥𝑥[𝑘𝑘]𝛿𝛿[𝑛𝑛 − 𝑘𝑘]+∞

𝑘𝑘=−∞

• LTI system can be represented by using unit impulse response.

• The output of LTI system is the convolution sum of input and unit impulse response.

• 2. Convolution sum

• 2. Convolution sum Computing method 1-- graphic method ▫ Step 1: change variable n k x1[n] x1[k], x2[n] x2[k] ▫ Step 2: reflect: x2[k] x2[-k] ▫ Step 3: shift: x2[-k] x2[n-k] ▫ Step 4: multiply and sum:

• 2. Convolution sum Computing method 2-- the property of

Note: only suitable for limited length sequence.

The representation of Continuous-Time Signals

• Approximate a CT signal x(t) as a sum of shifted, scaled pulses • If

• then

• so

∫∞

−∞=

−=τ

ττδτ dtxtx )()()(Basic Signals Weights

The Convolution Integral Representation of LTI System

• For a LTI system with the response of h(t) to the unit impulse δ(t)

x(t) y(t) CT LTI System

)()( tht →δ —— Unit Impulse Response

Time-invariance allows

)()( ττδ −→− tht

• Considering the weighted integral of delayed impulse representation of x(t)

• So

∫∞

∞−

−= ττδτ dtxtx )()()(

∫∞

∞−

−= τττ dthxty )()()(

)()()()()( thtxdthxty ∗=−= ∫∞

∞−

τττ

1. A LTI system is completely characterized by its response to the unit impulse ---- h(t)

2. The response y(t) to an input CT signal x(t) of a LTI system is the convolution of h(t) and x(t)

• Convolution Integral

τττ

txtxty

)()()(

−⋅=

∫∫

∞−

• Convolution Integral

τττ

txtxty

)()()(

−⋅=

∫∫

∞−

• Method 1 – graphic method ▫ Step 1. Replace t with τ for signals x1(t) and x2(t), i.e. τ

is the independent variable ▫ Step 2. Obtain the time reversal of x2(τ) ▫ Step 3. For the output value at any specific time t, shift

x2(-τ) with offset t to obtain x2(t-τ) ▫ Step 4. Multiply the two sequences x1(τ) and x2t-τ)

obtained in Step 1 and Step 3, respectively, and integrate the resulting product from to

−∞=τ ∞=τ

• Method 2 -- exploit the property of δ(t)

Supplements -- convolution

Supplements –δ function 1. Definition 2. Odd-even property (奇偶性） 3. The Differential and Integration Property （微积分特性）

Supplements –δ function 4. The Shifting Property in time domain （时移特性） 5. Multiply 6. Sifting property(筛选特性） 7. Scale property（尺度特性）

Candidate sets of basic signal

• Time domain ][/)( nt δδ

• Frequency domain

Candidate sets of basic signal

sttj ee /ω

The signal is decomposed into a linear combination of elementary signals

• The basic signal shall satisfy: • Can represent a fairly broad class of useful signals with the

"linear combination" of the basic signals. • The response of the LTI system to the base signal should

be very simple, and the response of the system to any input signal may be conveniently represented by the response of the base signals.

Complex exponential signal as basic signal

1. Representation

2. Response of LTI

stetx =)( τττττττ ττ dehedehdtxhty sstts −∞

∞−

−∞

∞−∫∫∫ ==−= )()()()()( )(

Let ττ τ dehsH s−∞

∞−∫= )()( ——System function,

assuming it is converged

stesHty )()( =The response to complex exponential of the LTI system:

stst esHe )(→

nn zzHz )(→Similarly, for DT systems, one obtains

the complex exponential with the same frequency, but scaled with H(s)

The response of a LTI system to a complex exponential is a complex exponential with the same frequency, but scaled with H(s)/H(z).

eigenvalue eigenfunction

nst ze /

)(/)( zHsH

—— eigenfunction (特征函数)

—— eigenvalue (特征值)

stst esHe )(→ nn zzHz )(→

∑∑ =→=k

kk esHatyeatx )()()(

∑∑ =→=k

nkk zzHanyzanx ))((][)(][

Following Eigenfunction property and superposition property of LTI systems, one obtains:

If the input to an LTI system is represented as a linear combination of complex exponentials, the output will be the linear combination of complex exponentials, each part is weighted by H(sk)/H(zk), i.e. the weighted value is depending on the frequency response associated with of the exponential component (sk/zk) .

Periodical complex exponential signal as basic signal

• Decompose signal as a linear combination of , and find out the response of the signal based on the response of .

------ The Fourier Transform

The Fourier transform of a continue time periodic signal is that the continue time periodic signal is represented by a linear combination of a group of harmonic signals or sinusoidal signals. Mathematically, they are a complete set of orthogonal functions.

Fourier Series Representation of CT Periodic Signals

-smallest such T is the fundamental period - is the fundamental frequency

Question #1: How do we find the Fourier coefficients?

multiply by Integrate over one period

denotes integral over any interval of length Here Next, note that

Orthogonality

CT Fourier Series Pair

(Synthesis equation)

(Analysis equation)

a )(10

—constant component or DC component of x(t)

0( ) j tx t e ω=

1 10,k 1

== ≠k

Example：

Fourier Series Representation of CT Periodic Signals: (1) The periodic signal x(t) could be constructed as a linear combination of

the harmonically related complex exponentials (sinusoidal signals) (2)ak is represents the magnitude and phase of kth harmonic component (3) {ak} are called as Fourier series coefficients, or spectrum of x(t) { | ak |} -- magnitude spectrum, {arg(ak)} --phase spectrum

Example：

)cos()( 0ttx ω=ka

21)cos()(

±≠=

eettx ωωω

Example:

)sin()( 0ttx ω=How about：

Magnitude Spectrum

Spectrum of x(t): Magnitude Spectrum, Phase Spectrum

Observations: 1. The spectrum of the periodic signal x(t) is discrete, it has non-zero values only at kω0, i.e. the spectrum space ω0 is 2π/T 2. ak is complex representing the magnitude and phase of kth harmonic component Notes: The negative frequencies (k<0) are meaningless in real world, they are for mathematical representations and derivations.

1)(1 2/

/2 == ∫−− πδ

∑+∞

−∞=

kTttx )()( δExample:

T 2T -T

… …

1 2 -1

… …

Ex: Periodic Square Wave

0=k ∫−

T TTdt

0≠k ∫−

− ==⋅=1

0)sin(

11 10T

tjkk k

πωω

——DC

1T1T−2T TT−

)(2)sin(10

110 TkSaTT

kTkak ω

xxxSa )sin()( =

0 4 8-4-8

where the sampling function is defined as

F.C. of the periodic square wave with fundamental frequency of T and pulse width of 2T1:

The envelop is the sampling function

Supposed T=8T1

The first zero value of the F.C. of the periodic square wave with fundamental frequency of T and pulse width of 2T1 is at the kth point satisfying kω0T1 =π, i.e. the main lobe of the signal is T/(2T1) (Hz), i.e. the bandwidth of the signal

Convergence of CT Fourier Series • The key is: What do we mean by

• One useful notion for engineers: there is no energy in the difference

(just need x(t) to have finite energy per period)

Under a different, but reasonable set of conditions (the Dirichlet conditions)

Condition 1. x(t) is absolutely integrable over one period, i. e.

And Condition 2. In a finite time interval, x(t) has

a finite number of maxima and minima. Ex. An example that violates

Condition 2.

And Condition 3. In a finite time interval, x(t) has only

a finite number of discontinuities. Ex. An example that violates Condition 3.

• Signals do not satisfy the Dirichlet conditions are generally pathological in nature, and do not typically exist in real world.

• • Dirichlet conditions are met for the signals we will encounter in the real world. Then - The Fourier series = x(t) at points where x(t) is continuous - The Fourier series = “midpoint” at points of discontinuity - There has no energy difference between the original signal and its

Fourier series representation - Since the original signal and its Fourier series representation only

differ at isolated points, the integral of both signals over any interval are identical, i.e. the two signals behave identically under convolution, and during the analysis of LTI systems.

• Still, convergence has some interesting characteristics:

-There exists error between the original signal x(t) and the approximation xN(t), it decreases as N increases - As N→ ∞, xN(t) exhibits Gibbs’ phenomenon at points of discontinuity (1.09)

katx ↔)( kbty ↔)(

•Linearity

kkk BbAaCtBytAxtz +=↔+= )()()(

•Time Shifting

ktjk aettx 00)( 0

ω−↔−

katx ↔)(

Time shifting introduces a linear phase shift ∝t0 in frequency domain

Example: Periodic Impulse Train

Example: Shift by half period

• Freuency Shifting

katx ↔)(

Eg. Carrier Modulation

MktjM aetx −↔0)( ω

•Time Reversal

katx −↔− )(

•Time Scaling

katx ↔)(α

katx ↔)(

∑∑ ==k

tjkk eatxeatx )( 00 )()( αωω α

katx ↔)(

Compression of a signal in time domain results in spectrum expand

Although the F.C. of x(at) and x(t) are identical, they have different fundamental frequency

-1/2 1 2 -1 -2

Example 3.6,p206

1 -1 2 3 4 5

21)1()( −−= txtg

•Multiplication

∑∞

−∞=−↔⋅

llklbatytx )()(

katx ↔)(

•Differential Integral

kajkdt

)( ω↔ k

ττ ↔∫∞−

katx ↔)(

0)(: 0=∫∞−

aifonlyperidoicandvaluedfiniteisdxNotet

kbty ↔)(

-2 2 4 8 6 0 -6

Example:

x’(t)

x”(t)

•Conjugation and Conjugate Symmetry

∗−

∗ ↔ katx )(

−∠=∠

}Im{}Im{}Re{}Re{

∗−= kk aa

katx ↔)(

If x(t) is real --Conjugate Symmetry

1( ) [ ( ) ( )] Re{ }21( ) [ ( ) ( )] Im{ }2

x t x t x t a

x t x t x t j a

= + − ↔

= − − ↔

oddandimaginarypurelyaoddandrealtxevenandrealaevenandrealtx

↔↔

•Parseval’s Relation

∑∫∞

−∞=

adttxT

22 |||)(|1

2|| ka

dttxT T∫ 2|)(|1

_average power in one period of x(t)

_average power in the kth harmonic component

Energy is the same whether measured in the time-domain or the frequency-domain

Fourier Series Representation of DT Periodic Signals

• x[n] -periodic with fundamental period N, fundamental frequency

• Only e jωn which are periodic with period N will appear in the FS

• There are only N distinct signals of this form

• So we could just use only N distinct exponential sequences to represent a DT periodic signal

• However, it is often useful to allow the choice of N consecutive values of k to be arbitrary.

DT Fourier Series Representation

= Sum over any N consecutive values of k

— This is a finite series

- Fourier (series) coefficients

Questions:

1) What DT periodic signals have such a representation?

2) How do we find ak?

Answer to Question #1:

Any DT periodic signal has a Fourier series representation

N equations for N unknowns, a0, a1, …, a N-1

A More Direct Way to Solve for ak

Finite geometric series

otherwise

So, from

multiply both sides by and then

orthoronality

DT Fourier Series Pair

(Synthesis equation)

(Analysis equation)

Note:It is convenient to think of ak as being defined for all integers k. So:

1) ak+N= ak —Special property of DT Fourier Coefficients.

2) We only use N consecutive values of ak in the synthesis equation. (Since x[n] is periodic, it is specified by N numbers, either in the time or frequency domain)

Example #1: Sum of a pair of sinusoids

periodic with period

Fourier Series Representation of CT Periodic Signals

T1 kept fixed T increases

Motivating Example

Discrete frequency

points become

denser in ω as T

increases

• Then for periodic square wave, the spectrum of x(t), i.e. {ak}, are , the spectrum space is

• Then for square pulse, the spectrum X(jω) are , the spectrum space is , i.e. the complex exponentials occur at a continuum of frequencies

TkTkak

10 )sin(2ωω

=Tπω 2

ωω )sin(2 1T

020 →=

1.0 Introduction

1.1.1 Development

• To derive the spectrum for aperiodic signals x(t), we can approximate it by a periodic signal with infinite period T

)(~ tx

-T1 T1

…… ……

-T1 0 T1 T

)(~ tx)()(~lim txtx

∞→

Assuming (1) is converged, we define

• Thus 𝑥𝑥� 𝑡𝑡 = �𝑎𝑎𝑘𝑘𝑒𝑒𝑗𝑗𝑘𝑘𝜔𝜔0𝑡𝑡 = �

1𝑇𝑇

𝑘𝑘𝑘𝑘

𝑋𝑋(𝑗𝑗𝑘𝑘𝜔𝜔0)𝑒𝑒𝑗𝑗𝑘𝑘𝜔𝜔0𝑡𝑡

2𝜋𝜋� 𝑋𝑋(𝑗𝑗𝑘𝑘𝜔𝜔0)𝑒𝑒𝑗𝑗𝑘𝑘𝜔𝜔0𝑡𝑡𝜔𝜔0

𝑘𝑘=−∞

• When 𝑇𝑇 → ∞ 𝑥𝑥 𝑡𝑡 =

12𝜋𝜋

� 𝑋𝑋(𝑗𝑗𝜔𝜔)𝑒𝑒𝑗𝑗𝜔𝜔𝑡𝑡𝑑𝑑𝜔𝜔∞

−∞

𝑥𝑥 𝑡𝑡 = 1

2𝜋𝜋 ∫ 𝑋𝑋(𝑗𝑗𝜔𝜔)𝑒𝑒𝑗𝑗𝜔𝜔𝑡𝑡𝑑𝑑𝜔𝜔∞−∞

𝑋𝑋 𝑗𝑗𝜔𝜔 = ∫ 𝑥𝑥(𝑡𝑡)𝑒𝑒−𝑗𝑗𝜔𝜔𝑡𝑡𝑑𝑑𝑡𝑡∞−∞

Synthesis equation

Analysis equation

1.1.2 Convergence

• What kinds of signals can be represented in Fourier Transform (satisfies one of the following 2 conditions) ▫ 1、Finite energy

Then we are guaranteed that: 𝑋𝑋(𝑗𝑗𝜔𝜔) is finite ∫ 𝑒𝑒(𝑡𝑡) 2𝑑𝑑𝑡𝑡∞

−∞ = 0

(𝑒𝑒 𝑡𝑡 = 𝑥𝑥� 𝑡𝑡 − 𝑥𝑥(𝑡𝑡) 𝑥𝑥� 𝑡𝑡 = 12𝜋𝜋 ∫ 𝑋𝑋(𝑗𝑗𝜔𝜔)𝑒𝑒𝑗𝑗𝜔𝜔𝑡𝑡𝑑𝑑𝜔𝜔∞

−∞ )

▫ 2、Dirichlet conditions, require that 𝑥𝑥(𝑡𝑡) be absolutely integrable 𝑥𝑥(𝑡𝑡) have a finite number of maxima and minima

within any finite interval 𝑥𝑥(𝑡𝑡) have a finite number of discontinuities within

any finite interval. Furthermore, each of these discontinuities must be finite

Then we guarantee that 𝑥𝑥� 𝑡𝑡 is equal to 𝑥𝑥(𝑡𝑡) for any 𝑡𝑡 except at a

discontinuity, where it is equal to the average of the values on either side of the discontinuity

𝑋𝑋(𝑗𝑗𝜔𝜔) is finite

Examples • Exponential function

Magnitude Spectrum Phase Spectrum

Even symmetry Odd symmetry

If α is complex, x(t) is absolutelty

integrable as long as Re{α}>0

2)(0，)(

ωααωαα

+=↔>=

−jXetx

1.0 Introduction

• For a periodic signal x(t) with fundamental frequency , what’s its FT?

Tπω 2

tjkkeatx 0)( ω

∑∑ ℑ=ℑ=ℑ∴k

tjkk eaeatx ][][)]([ 00 ωω

?0 ↔tjke ωthe question becomes:

• Thanks to the impulse function, suppose 𝑋𝑋 𝑗𝑗𝜔𝜔 = 𝛿𝛿 𝜔𝜔 − 𝜔𝜔0

𝑥𝑥 𝑡𝑡 =1

2𝜋𝜋� 𝛿𝛿 𝜔𝜔 − 𝜔𝜔0 𝑒𝑒𝑗𝑗𝜔𝜔𝑡𝑡𝑑𝑑𝜔𝜔∞

−∞=

12𝜋𝜋

𝑒𝑒𝑗𝑗𝜔𝜔0𝑡𝑡

• That is 𝑒𝑒𝑗𝑗𝜔𝜔0𝑡𝑡 ↔ 2𝜋𝜋𝛿𝛿 𝜔𝜔 − 𝜔𝜔0

• So 𝑥𝑥 𝑡𝑡 = �𝑎𝑎𝑘𝑘𝑒𝑒𝑗𝑗𝑘𝑘𝜔𝜔0𝑡𝑡

𝑘𝑘

↔ 𝑋𝑋 𝑗𝑗𝜔𝜔 = �2𝜋𝜋𝑎𝑎𝑘𝑘𝑘𝑘

𝛿𝛿 𝜔𝜔 − 𝑘𝑘𝜔𝜔0

— All the energy is concentrated in one frequency — ωo,

• So for a periodic signal x(t) with fundamental frequency , its FT is: ▫ Fourier Series Coefficient ▫ Fourier Transform

• The FT can be interpreted as a train of impulses occurring at the harmonically related frequencies and for which the area of the impulse at the kth harmonic frequency kω0 is 2π times the kth F.S. coefficient ak

Tπω 2

dtetxT

0 02( ) ( 2 ( )k

kx t X j a k

Tπω π δ ω ω ω

=−∞

↔ = − =∑），

1.0 Introduction

• Linearity

• Time Shifting

)()( ωjXtx ↔ )()( ωjYty ↔

)()()()( ωω jbYjaXtbytax +↔+

)()( ωjXtx ↔

)()( 00 ωω jXettx tj−↔−

• Time and Frequency Scaling

)()( ωjXtx ↔

1)(ajX

aatx ω

1−=a )()( ωjXtx −↔−for

compressed in time ⇔ stretched in frequency

• Differentiation

• The differentiation operation enhances high-frequency components in the effective frequency band of a signal

• Without any further information about the DC component of the original signal, we cannot completely recover it from its differentials

)()( ωjXtx ↔

)()( ωω jXjdt

tdx↔

• Integration

ττ dxtgt

∫∞−

= )()( )()( ωjXtx ↔

)()0()(1)()()( ωδπωω

ωττ XjXj

jGdxtgt

+=↔= ∫∞−

The integration operation diminishes high-frequency components in the effective frequency band of a signal

• Duality ▫ Both time and frequency are continuous and in general aperiodic

▫ Suppose f() and g() are two functions related by

Same except for these differences

)()( ωjXtx ↔ )(2)( ωπ jxtX −↔

• Other duality properties ▫ （1） Frequency Shifting

)()( ωjXtx ↔

))(()( 00 ωωω −↔ jXtxe tj

▫ （2）Differentiation in frequency domain

▫ （3）Integration in frequency domain

djdXtjtx )()( ↔−

djdXjttx )()( ↔

λλδπω

dXtxtxjt ∫

∞−

↔+− )()()0()(1

∫∞−

−↔t

dXjttx λλ)()(

when x(0)=0,

• Conjugation and Conjugate Symmetry )()( ωjXtx ↔

)()( ωjXtx −↔ ∗∗

If x(t) is real valued

)()( ωω jXjX −= ∗ —Conjugate Symmetry

)]([)](Re[)( ωωω jXjIjXjX m+=

)](Re[)](Re[ ωω jXjX −=

)](Im[)](Im[ ωω jXjX −−=

)(|)(|)( ωωω jXjejXjX ∠=

|)(||)(| ωω jXjX −＝

)()( ωω jXjX −∠−∠ ＝

evenandrealtx )( evenandrealjX )( ω↔oddandrealtx )( oddandimaginarypurelyjX )( ω↔

)]([)]()([21)( ωjXRetxtxtxe ↔−+=

)]([)]()([21)(0 ωjXjItxtxtx m↔−−=

• Parseval’s Relation

dffXdjXdttx ∫∫∫∞

∞−

== 222 |)(||)(|21|)(| ωωπ

2|)(| ωjX

ωωπ

dTjXdttx

T ∫∫∞

∞−∞→∞→

2 |)(|lim21|)(|1lim

2|)(|lim ω∞→

——Energy-density Spectrum

——Power-density Spectrum

and：

2|)(| fX——Energy per unit frequency (Hz)

1.0 Introduction

1.4.1 Convolution Property

1.4.2 Frequency Response

• Definition:

∞<∫∞

∞−

dtth )( －stable system

( )( ) ( ) ( ) / ( )( )

Y jY j X j H j H jX j

ωω ω ω ωω

Conditioned on:

1.4.3 Filtering －a process in which the relative complex magnitudes of the frequency components in a signal are changed or some frequency components are completely eliminated • Frequency-Selective Filters —systems that are designed to pass some frequency components undistorted, and diminish/eliminate others significantly • Typical types of frequency-selective filters

▫ LPF(Low-pass Filter) ▫ HPF(High-pass Filter) ▫ BPF(Band-pass Filter) ▫ BSF (Band-stop Filter

)( ωjH

cωcω−

cjHωωωω

，LPF

passband stopband

cjHωωωω

<<≥≤

ccjHωωωωωωω

ω，，

cutoff frequency

upper cutoff frequency lower cutoff frequency

• Time domain and frequency domain aspects of non-ideal filter

▫ Trade-offs between time domain and frequency domain characteristics, i.e. the

width of transition band ↔ the setting time of the step response

Definitions: Passband ripple: δ1 Stopband ripple: δ2

Definitions: Rise time: tr Setting time: ts Overshoot: Δ Ringing frequency ωr

Passband Transition Stopband Rise time

Setting time

Setting time: the time at which the step response settles to within δ (a specified tolerance) of its final value

1.0 Introduction

1.5.1 Multiplication Property

)()()()( 2211 ωω jXtxjXtx ↔↔

)()()( 21 txtxtx ⋅=

)]()([21)( 21 ωωπ

ω jXjXjX ∗=

1.5.2 Sampling • Most of the signals we encounter are CT signals, e.g. x(t). How do we

convert them into DT signals x[n] to take advantages of the rapid progress and tools of digital signal processing ▫ — Sampling, taking snap shots of x(t) every T seconds

• T –sampling period, x[n] ≡x(nT), n= ..., -1, 0, 1, 2, ... — Regularly spaced samples

• Applications and Examples ▫ —Digital Processing of Signals ▫ —Images in Newspapers ▫ —Sampling Oscilloscope ▫ —… How do we perform sampling?

• Why/When Would a Set of Samples Be Adequate? ▫ Observation: Lots of signals have the same samples

▫ By sampling we throw out lots of information –all values of x(t) between sampling points are lost.

▫ Key Question for Sampling: Under what conditions can we reconstruct the original CT signal x(t) from its samples?

• Impulse Sampling—Multiplying x(t) by the sampling function

• Analysis of Sampling in the Frequency Domain

Multiplication Property =>

=Sampling Frequency

• Illustration of sampling in the frequency-domain for a band-limited (X(jω)=0 for |ω| > ωM) signal

• Reconstruction of x(t) from sampled signals

If there is no overlap

between shifted

spectra, a LPF can

reproduce x(t) from xp(t)

Suppose x(t) is band-limited, so that

X(jω)=0 for |ω| > ωM

Then x(t) is uniquely determined by its

samples {x(nT)} if

where ωs = 2π/T

• Observations ▫ (1) In practice, we obviously don’t

sample with impulses or implement ideal low-pass filters

— One practical example: The Zero-Order Hold ▫ (2) Sampling is fundamentally a

time varying operation, since we multiply x(t) with a time-varying function p(t). However, H(jω) is the identity system (which is TI) for band-limited x(t) satisfying the sampling theorem (ωs > 2ωM).

▫ (3) What if ωs <= 2ωM? Something different: more later.

,...,2,1,0),()(.0)(

txequalexactlywillsignaloutputresultingthe

iffrequencycutoffandTgainwithfilterlowpassidealanthroughprocessedthenistrainimpulseThisvaluessamplesuccessive

arethatamplitudeshaveimpulsesuccessivewhichintrainimpulseperiodicageneratingbytxtreconstruccanwesamplestheseGiven

Twhere

ifnnTxsamplesitsbydetermineduniquelyistxThenforjXwithsignallimitedbandabeusLet

TheoremSampling

ωωωωω

ωωω

>±±=

• Time-Domain Interpretation of Reconstruction of Sampled Signals —Band-Limited Interpolation

The lowpass filter interpolates the samples assuming x(t) contains no

energy at frequencies >= ωc

• Graphic Illustration of Time-Domain Interpolation

▫ Original CT signal

▫ After Sampling

▫ After passing the LPF

Many Thanks

Chapter 1 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/wavelet/continuous time...

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