Chapter 2

Updated 04/20/23

Outline

• Transformation of Continuous-Time Signal– Time Reversal– Time Scaling– Time Shifting– Amplitude Transformation

• Signal Characteristics

Time reversal:

Time Reversal

X(t) Y=X(-t)

Mathematica Example

Shift+<Enter> to execute

Time scaling

TimeScaling

X(t) Y=X(at)

Time scaling

• Given y(t), – find w(t) = y(3t) – v(t) = y(t/3).

Circuit Example

• LC Tank Oscillator

Time Shifting

• The original signal x(t) is shifted by an amount to .Time Shift: y(t)=x(t-to)

• X(t)→X(t-to) // to>0 → Signal Delayed → Shift to the right

• X(t) → X(t+to) // to<0 → Signal Advanced → Shift to the left

TimeShifting

X(t) Y=X(t-to)

Connection to Circuits

Note: Unit Step Function

Unit Step function(a discontinuous continuous-time signal):

Mathematica Example (1)

Mathematica Example (2)

Draw

• x(t) = u(t+1)- u(t-2)

u(t+1)- u(t-2)

t=0

Mathematica Example (2)

Time Shifting Example

• Given x(t) = u(t+2) -u(t-2), – find

• x(t-t0)=• x(t+t0)=Answer:• x(t-t0)= u(t-to+2) -u(t-to-2), • x(t+t0)= u(t+to+2) -u(t+to-

2),

Problem

• Determine x(t) + x(2-t) , where x(t) = u(t+1)- u(t-2

• Method 1:– Observation: Rewrite x(2-t) as x(-(t-2))– Find x(-t) first, then shift t by t-2.

• Method 2: – Observation: X(2-t) implies time reversal.– So find x(2+t), then apply time reversal

Method 1

Find x(-t) first, then shift t by t-2.

Method 2

find x(2+t), then apply time reversal

X(2-t)+x(t)

X(2-t)

X(t)

X(2-t)+x(t)

Combination of Scaling and Shifting

Method 1: Shift then scale

Combination of Scaling and Shifting

Method 2: Scale then shift

Amplitude Operations

In general: y(t)=Ax(t)+B

B>0 Shift upB<0 Shift down

|A|>1 Gain |A|<1 Attenuation

A>0NO reversal A<0 reversal

Reversal

Scaling

Scaling

Y(t)=AX(t)+B Example

Input and Output

Vout, m=46 mVVin, m=1 mV

Define a Piecewise Function in Mathematica

Example 2-1

X(t)

Advance: X(t+1)

Advance & ScalingX(t/2+1)

Advance,scaling &reversalX(-t/2+1)

Signal Characteristics

• Even Function

Xe(-t) = Xe(t)

Signal Characteristics

• Odd Function

Xo(t) =- Xo(-t)

Signal Characteristics

Xe + Ye = ZeXo + Yo = ZoXe + Yo = Ze + Zo

Xe * Ye = ZeXo * Yo = ZeXe * Yo = Zo

Any signal can be represented in terms of a odd function and an even function.

x(t)=xo(t)+xe(t)

Represent xe(t) in terms of x(t)

• Xe(t)

– X(t)=Xe(t)+Xo(t)

– Xe(t)=X(t)+Xo(t)• Xo(t)=-Xo(-t)

• X(-t)=Xe(-t)+Xo(-t)

– Xe(t)=X(t)-Xo(-t)=X(t)+X(-t)-Xe(-t)

• Therefore Xe(t)=[X(t)+X(-t)]/2• Similarly Xo(t)=[X(t)-X(-t)]/2

Proof Examples• Prove that product of two

even signals is even.

• Prove that product of two odd signals is even.

• What is the product of an even signal and an odd signal? Prove it!

)()()(

)()()(

)()()(

21

21

21

txtxtx

txtxtx

txtxtx

Oddtx

txtxtx

txtxtx

txtxtx

)(

)()()(

)()()(

)()()(

21

21

21

Change t -t

(even) (odd)