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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.

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

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.

Sum of sinusoids

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 ) 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.

For s in the shaded region, the response to x ( t ) = Xe st size 12{x \( t \) = ital "Xe" rSup { size 8{ ital "st"} } } {} is the steady-state response y ( t ) = Ye st = XH ( s ) e st 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)|.

2/ Effect of a zero

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 ) 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 s k ) size 12{ \( s - s rSub { size 8{k} } \) } {} . Each of these terms are vectors in the complex s plane:

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

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

III. FREQUENCY RESPONSE

1/ Relation to system function

Note if

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 ( ) e jωt + XH ( ) e jωt 2 y ( t ) = XH ( ) e jωt 2 + XH ( ) e jωt 2 y ( t ) = 2R XH ( ) e jωt 2 y ( t ) = 2R X H ( ) e j arg ( H ( ) ) e jωt y ( t ) = X H ( ) cos ( ωt + arg ( H ( ) ) ) alignl { stack { size 12{y \( t \) = { { ital "XH" \( jω \) e rSup { size 8{jωt} } + ital "XH" \( - jω \) e rSup { size 8{ - jωt} } } over {2} } } {} #y \( t \) = { { ital "XH" \( jω \) e rSup { size 8{jωt} } } over {2} } + left [ { { ital "XH" \( jω \) e rSup { size 8{jωt} } } over {2} } right ] rSup { size 8{*} } {} #y \( t \) =2R left lbrace { { ital "XH" \( jω \) e rSup { size 8{jωt} } } over {2} } right rbrace {} # y \( t \) =2R left lbrace X \lline H \( jω \) \lline e rSup { size 8{j"arg" \( H \( jω \) \) } } e rSup { size 8{jωt} } right rbrace {} #y \( t \) =X \lline H \( jω \) \lline "cos" \( ωt+"arg" \( H \( jω \) \) \) {} } } {}

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Source:  OpenStax, Signals and systems. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10803/1.1
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