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Lecture 11:Sinusoidal Steady-State (SSS) Or Frequency Response

Module by: Thanh Dinh Vu, Truc Pham-Dinh, Anh Tuan Hoang, Tam Huynh-Ngoc

Summary: SSS response essentially characterizes the system function. SSS response is commonly used to measure the system function. A common mode of thinking for signal processing tasks is “filtering.” All physical systems act as filters of their input signals. Filtering is an important signal processing method.

Links

Supplemental links

[3] BIBO Stability
[3] Frequency Response
[3] Signal Processing Basics
[3] Signal Processing
[3] Introduction to Fundamentals of Signal Processing

Lecture #11:

SINUSOIDAL STEADY-STATE (SSS) OR FREQUENCY RESPONSE

Motivation:

Many systems operate in the SSS, e.g., electrical power distribution, broadcast, touch-tone telephone.

SSS response essentially characterizes the system function.

SSS response is commonly used to measure the system function.

A common mode of thinking for signal processing tasks is “filtering.” All physical systems act as filters of their input signals. Filtering is an important signal processing method.

Outline:

Causal, stable systems
Sinusoidal steady-state response—the frequency response
Relation of frequency response to system function
Bode diagrams

Signal processing with filters
Lowpass and highpass filters
Resonance and bandpass filters
Notch filters
Conclusions

A system that operates in the SSS . . . well almost

The touch-tone phone

Figure 1

Dialing consists of pressing buttons on the keypad which has 3 columns and 4 rows. How is the information about which button is pushed coded?

Demo on coding in the touch-tone phone.

Figure 2

Figure 3

Sum of sinusoids

s ( t ) = sin ( 2πf 1 t ) + sin ( 2πf 2 t )

s ( t ) = sin ( 2πf 1 t ) + sin ( 2πf 2 t ) size 12{s \( t \) ="sin" \( 2πf rSub { size 8{1} } t \) +"sin" \( 2πf rSub { size 8{2} } t \) } {}

times a pulse window p(t)

x ( t ) = p ( t ) s ( t )

x ( t ) = p ( t ) s ( t ) size 12{x \( t \) =p \( t \) s \( t \) } {}

More on the effect of p(t) later!

I. CAUSAL, STABLE SYSTEMS

The system function of an LTI system, H(s), can be used to categorize systems.

Causal system ⇒⇒ size 12{ drarrow } {} ROC is to the right of the rightmost pole of H(s).

Causal, stable system ⇒⇒ size 12{ drarrow } {} ROC is to the right of the rightmost pole of H(s) and all the poles are in the left-half of the s plane.

More on the definition of stable systems at a later time.

A causal, stable system has the pole-zero diagram shown below.

Figure 4

For s in the shaded region, the response to x(t)=Xest

x(t)=Xest size 12{x \( t \) = ital "Xe" rSup { size 8{ ital "st"} } } {}

is the steady-state response y(t)=Yest=XH(s)est

y(t)=Yest=XH(s)est size 12{y \( t \) = ital "Ye" rSup { size 8{ ital "st"} } = ital "XH" \( s \) e rSup { size 8{ ital "st"} } } {}

II. THE SYSTEM FUNCTION H(S)

1/ Real and complex poles

H(s) is a complex function of a complex variable s. The plots show |H(s)|.

Figure 5

2/ Effect of a zero

Figure 6

3/ Interpretation of H(s) by pole and zero vectors

H(s) that is a rational function has the form

H ( s ) = K ( s − z 1 ) ( s − z 2 ) . . . ( s − z M ) ( s − p 1 ) ( s − p 2 ) . . . ( s − p N )

H ( s ) = K ( s − z 1 ) ( s − z 2 ) . . . ( s − z M ) ( s − p 1 ) ( s − p 2 ) . . . ( s − p N ) size 12{H \( s \) =K { { \( s - z rSub { size 8{1} } \) \( s - z rSub { size 8{2} } \) "." "." "." \( s - z rSub { size 8{M} } \) } over { \( s - p rSub { size 8{1} } \) \( s - p rSub { size 8{2} } \) "." "." "." \( s - p rSub { size 8{N} } \) } } } {}

H(s) consists of products and quotients of the form (s−sk)

(s−sk) size 12{ \( s - s rSub { size 8{k} } \) } {}

. Each of these terms are vectors in the complex s plane:

(s−z1)(s−z2)...(s−zM)

(s−z1)(s−z2)...(s−zM) size 12{ \( s - z rSub { size 8{1} } \) \( s - z rSub { size 8{2} } \) "." "." "." \( s - z rSub { size 8{M} } \) } {}

are called zero vectors,

(s−p1)(s−p2)...(s−pN)

(s−p1)(s−p2)...(s−pN) size 12{ \( s - p rSub { size 8{1} } \) \( s - p rSub { size 8{2} } \) "." "." "." \( s - p rSub { size 8{N} } \) } {}

are called pole vectors.

The vector (s−sk)

(s−sk) size 12{ \( s - s rSub { size 8{k} } \) } {}

points from sk

sk size 12{s rSub { size 8{k} } } {}

to s. It can be expressed in polar form as

( s − s k ) = ∣ s − s k ∣ e j arg ( s − s k )

( s − s k ) = ∣ s − s k ∣ e j arg ( s − s k ) size 12{ \( s - s rSub { size 8{k} } \) = \lline s - s rSub { size 8{k} } \lline e rSup { size 8{j"arg" \( s - s rSub { size 6{k} } \) } } } {}

We now take

H ( s ) = K ( s − z 1 ) ( s − z 2 ) . . . ( s − z M ) ( s − p 1 ) ( s − p 2 ) . . . ( s − p N )

and express the vectors in polar form

{}

H ( s ) = K ∣ s − z 1 ∣ e j arg ( s − z 1 ) ∣ s − z 2 ∣ e j arg ( s − z 2 ) . . . ∣ s − z M ∣ e j arg ( s − z M ) ∣ s − p 1 ∣ e j arg ( s − p 1 ) ∣ s − p 2 ∣ e j arg ( s − p 2 ) . . . ∣ s − p N ∣ e j arg ( s − p N )

H ( s ) = K ∣ s − z 1 ∣ e j arg ( s − z 1 ) ∣ s − z 2 ∣ e j arg ( s − z 2 ) . . . ∣ s − z M ∣ e j arg ( s − z M ) ∣ s − p 1 ∣ e j arg ( s − p 1 ) ∣ s − p 2 ∣ e j arg ( s − p 2 ) . . . ∣ s − p N ∣ e j arg ( s − p N ) size 12{H \( s \) =K { { \lline s - z rSub { size 8{1} } \lline e rSup { size 8{j"arg" \( s - z rSub { size 6{1} } \) } } \lline s - z rSub {2} size 12{ \lline e rSup {j"arg" \( s - z rSub { size 6{2} } \) } } size 12{ "." "." "." \lline s - z rSub {M} } size 12{ \lline e rSup {j"arg" \( s - z rSub { size 6{M} } \) } }} over { size 12{ \lline s - p rSub {1} size 12{ \lline e rSup {j"arg" \( s - p rSub { size 6{1} } \) } } size 12{ \lline s - p rSub {2} } size 12{ \lline e rSup {j"arg" \( s - p rSub { size 6{2} } \) } } size 12{ "." "." "." \lline s - p rSub {N} } size 12{ \lline e rSup {j"arg" \( s - p rSub { size 6{N} } \) } }} } } } {}

so that

∣ H ( s ) ∣ = ∣ K ∣ ∣ s − z 1 ∣ ∣ s − z 2 ∣ . . . ∣ s − z M ∣ ∣ s − p 1 ∣ ∣ s − p 2 ∣ . . . ∣ s − p M ∣

∣ H ( s ) ∣ = ∣ K ∣ ∣ s − z 1 ∣ ∣ s − z 2 ∣ . . . ∣ s − z M ∣ ∣ s − p 1 ∣ ∣ s − p 2 ∣ . . . ∣ s − p M ∣ size 12{ \lline H \( s \) \lline = \lline K \lline { { \lline s - z rSub { size 8{1} } \lline \lline s - z rSub { size 8{2} } \lline "." "." "." \lline s - z rSub { size 8{M} } \lline } over { \lline s - p rSub { size 8{1} } \lline \lline s - p rSub { size 8{2} } \lline "." "." "." \lline s - p rSub { size 8{M} } \lline } } } {}

And

arg H ( s ) = arg K + arg ( s − z 1 ) + arg ( s − z 2 ) + . . . + arg ( s − z M ) − arg ( s − p 1 ) − arg ( s − p 2 ) − . . . − arg ( s − p N )

arg H ( s ) = arg K + arg ( s − z 1 ) + arg ( s − z 2 ) + . . . + arg ( s − z M ) − arg ( s − p 1 ) − arg ( s − p 2 ) − . . . − arg ( s − p N ) size 12{"arg"H \( s \) ="arg"K+"arg" \( s - z rSub { size 8{1} } \) +"arg" \( s - z rSub { size 8{2} } \) + "." "." "." +"arg" \( s - z rSub { size 8{M} } \) - "arg" \( s - p rSub { size 8{1} } \) - "arg" \( s - p rSub { size 8{2} } \) - "." "." "." - "arg" \( s - p rSub { size 8{N} } \) } {}

And

H ( s ) = ∣ H ( s ) ∣ e j arg ( H ( s ) )

H ( s ) = ∣ H ( s ) ∣ e j arg ( H ( s ) ) size 12{H \( s \) = \lline H \( s \) \lline e rSup { size 8{j"arg" \( H \( s \) \) } } } {}

III. FREQUENCY RESPONSE

1/ Relation to system function

Figure 7

Figure 8

Note if

x ( t ) = Xe jωt + Xe − jωt 2 = X cos ( ωt )

x ( t ) = Xe jωt + Xe − jωt 2 = X cos ( ωt ) size 12{x \( t \) = { { ital "Xe" rSup { size 8{jωt} } + ital "Xe" rSup { size 8{ - jωt} } } over {2} } =X"cos" \( ωt \) } {}

then

y ( t ) = XH ( jω ) e jωt + XH ( − jω ) e − jωt 2 y ( t ) = XH ( jω ) e jωt 2 + XH ( jω ) e jωt 2 y ( t ) = 2R XH ( jω ) e jωt 2 y ( t ) = 2R X ∣ H ( jω ) ∣ e j arg ( H ( jω ) ) e jωt y ( t ) = X ∣ H ( jω ) ∣ cos ( ωt + arg ( H ( jω ) ) )

y ( t ) = XH ( jω ) e