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Chapter 4/5 CT Fourier Transform DT Fourier Transform For non-periodic signal

() = ()

= []=

is periodic in with period 2

() = 12 () [] = 12 2 For periodic signal () = 2( 0)

=

0 = 2 = 2( 0)

=

0 = 2 Properties Table 4.1 Table 5.1 Basic Transform Pairs

Table 4.2 Table 5.2

Chapter 9/10 Laplace Transform Z Transform

() = ()

() = []=

= + () = ()|=0

() = {()} = = ()|=1() = {[]} Properties Table 9.1 Table 10.1 Basic Transform Pairs

Table 9.2 Table 10.2

ROC ROC Finite Duration Signal

If x(t) is absolutely integrable, entire s-plane

Entire z-plane except possibly z = 0 and/or z =

Right -sided Signal

If () = ()() (rational), right half

plane to the right of the right-most pole

If () = ()() (rational), outside the circle of

the outmost pole (special care for z = )

Left-sided signal

If () = ()() (rational), left half plane

to the left of the left-most pole

If () = ()() (rational), inside the circle of

the innermost nonzero pole (special care for z = 0)

Two-sided signal

If () = ()() (rational), the strip

parallel to the j axis bounded by poles

If () = ()() (rational), the ring bounded by

poles ()

() , . ., |()|

< ({} = 0)

[] , . ., |[]|

=

< (|| = 1)

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LTI System LTI

h(t)x(t) y(t)

LTIh[n]

x[n] y[n]

Impulse response

h(t)

h[n]

() = () ()= ()( )

[] = [] []= [][ ]=

Frequency response

H(j) H(ej)

() = ()() = () System or Transfer function

H(s) H(z)

() = ()() () = ()() Properties Causal If () = ()

() (rational), ROC of a causal system function is the right half plane to the right of the right-most pole.

If () = ()() (rational), ROC of a causal

system function is the exterior of a circle outside of the outmost pole and the order of the numerator cannot be greater than the order of the denominator.

Stable ROC of a stable system function includes the entire j axis ({} = 0). ROC of a stable system function includes the unit circle (|z|=1).

Causal and Stable

ROC of a causal and stable system function has all poles in the left-half plane, i.e., all of the poles have negative real parts.

ROC of a causal and stable system function has all poles inside the unit circle, i.e., all of the poles have magnitude smaller than 1.

CT LTI system characterized by linear-

constant coefficient differential equation

DT LTI system characterized by linear-constant coefficient difference equation

()

=0

= () =0

[ ]=0

= [ ]=0

Frequency Response () ()

=0

= () ()=0

=0

= =0

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() = ()() = ()=0 [ ()=0 ]

= () = =0 [ =0 ]

System or Transfer Function

() =0

= () =0

() = ()() = =0 [ =0 ]

() =0

= () =0

() = ()() = =0 [ =0 ]

Interconnections of LTI system

Serial h1(t) h2(t)x(t) y(t)

h1[n] h2[n]x[n] y[n]

() = 1() 2() () = 1()2() () = 1()2()

[] = 1[] 2[] = 12 () = 1()2()

Parallel

+x(t) y(t)

h1(t)

h2(t)

+x[n] y[n]

h1[n]

h2[n]

() = 1() + 2() () = 1()+2() () = 1()+2()

[] = 1[] + 2[] = 1+2 () = 1()+2()

Chapter 7 Sampling

Sampling Theorem

If > 2, we can use [] =() to reconstruct () perfectly. : sampling frequency : highest frequency in signal 2: Nyquist rate

If 2, aliasing occurs.

Impulse Train Sampling

() = ()() () = ( )= , = 2. () is a periodic signal with Fourier series coefficient = 1. () = 1

2( )=

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() = 12() () = 1 ( )=

1X(j)

-M

1/Ts

Xp(j)

-s

... ...

M -M M s

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