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The System Function and The Frequency Response of LTI Systems (Review)
Recall,
and the output of an LTI system may be expressed in the -domain as
For a special class of LTI systems that can be described by a linear, constant-coefficient difference equation of the form
the system function is
When ,
and
The system has a finite-duration impulse response (FIR), and its frequency response is composed of all zeros (and poles at the origin).
When , the system is called an infinite-duration impulse
response (IIR) system.
H z ( ) h n ( ) z – n n =–∞
∞
z Y z ( ) = H z ( ) X z ( )
y n ( ) a (^) k y n ( – k ) k = 1
N
– ∑ b k x n ( – k )
k = 0
M
H z ( ) Y z ( ) X z ( )
b (^) k z
k = 0
M
1 a (^) k z – k k = 1
N
{ a (^) k } = 0 H z ( ) b (^) k z – k k = 0
M
h n ( )
b (^) n ,
0,
0 ≤ n ≤ M otherwise
a (^) k ≠ 0
The Frequency Response Function
The Fourier transform relationship between the impulse response and the frequency response function is given by:
Recall that this function is periodic with period. The output of an LTI
system with frequency response to an aperiodic finite energy signal
with Fourier transform is given as:
The frequency response function is usually expressed in terms of its
magnitude and its phase , where
usually, the magnitude is plotted on a logarithmic scale as
where the units are decibels (dB). Sometimes we normalize so that its maximum value is unity (zero on the dB scale). Othertimes, we normalize
so that its energy is equal to unity.
Example:
H (ω ) h n ( ) e – j ω n n =–∞
∞
2 π H (ω ) X (ω ) Y (ω ) = H (ω ) X (ω )
H (ω ) Θ ω( )
H (ω ) H (ω ) e
j Θ ω( )
H (ω ) (^) dB 20 log 10 H (ω ) 10 H (ω )
2 = = log 10
H (ω )
H (ω )
y n ( ) = 1.8 y n ( – 1 ) – 0.81 y n ( – 2 ) + x n ( ) +0.95 x n ( – 1 )
H (ω ) H (ω ) (^) max
log -------------------------
1 +0.95 e – j ω
1 – 1.8 e – j ω+0.81 e –^2 j ω
Hence, when is real, the complex-valued poles and zeros occur in
complex-conjugate pairs, and , or , so
If , and the transforms , and
are the -transforms of the autocorrelation sequences and ,
where
we can show that may be expressed as a ratio of polynomial
functions of :
Note that
h n ( )
H *^ ( 1 ⁄ z *) = H z ( –^1 ) H *^ (ω ) = H (– ω)
H (ω )
2 H (ω ) H *^ (ω ) H (ω ) H (– ω) H z ( ) H^1 z
z = e j ω
H z ( ) B z ( ) A z ( )
= ---------- D z ( ) B z ( ) B^1 z
= ( ) --- C z ( ) A z ( ) A^1 z
z { c (^) l } { d (^) l }
c (^) l a (^) k a (^) k + l k = 0
N – l
= ∑ , – N ≤ l ≤ N
d (^) l b (^) k b (^) k + l k = 0
M – l
= ∑ , – M ≤ l ≤ M
H (ω ) 2 cos ω
H (ω ) 2
d 0 2 d (^) k cos k ω k = 1
M
c 0 2 c (^) k cos k ω k = 1
N
cos k ω β m ( cosω) m m = 0
k
Interconnections of LTI Systems
Recall for cascaded systems:
For parallel systems, no such simple relationships hold.
Example:
H (ω ) = H 1 (ω^ ) H 2 (ω^ ) 20 log 10 H (ω ) = 20 log 10 H 1 (ω ) + 20 log 10 H 2 (ω )
Θ ω( ) = Θ 1 (ω ) +Θ 2 (ω )
h 4 ( ) n
h 1 ( ) n
h 3 ( ) n
h 2 ( ) n x n ( ) y n ( )
System Function:
H z ( ) = H 4 ( ) z + H 1 ( ) z [ H 2 ( ) z + H 3 ( ) z ]
