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λ 1,2 = ( 1 + rg 2 ) ± ( 1 + rg 2 ) 2 1 size 12{λ rSub { size 8{1,2} } = \( 1+ { { ital "rg"} over {2} } \) +- sqrt { \( 1+ { { ital "rg"} over {2} } \) rSup { size 8{2} } - 1} } {}

The following plot shows the locus of natural frequencies as rg increases.

Both natural frequencies lie along the positive, real λ-axis. When rg = 0, λ 1,2 size 12{λ rSub { size 8{1,2} } } {} = 1. As rg increases, one natural frequency decreases toward λ = 0 and the other natural frequency increases. What is the physical significance of this pattern?

Note that the product of the two natural frequencies is 1,

λ 1 λ 2 = ( 1 + rg 2 + ( 1 + rg 2 ) 2 1 ) × ( 1 + rg 2 ( 1 + rg 2 ) 2 1 ) = 1 size 12{λ rSub { size 8{1} } λ rSub { size 8{2} } = \( 1+ { { ital "rg"} over {2} } + sqrt { \( 1+ { { ital "rg"} over {2} } \) rSup { size 8{2} } - 1} \) times \( 1+ { { ital "rg"} over {2} } - sqrt { \( 1+ { { ital "rg"} over {2} } \) rSup { size 8{2} } - 1} \) =1} {}

Therefore, λ 1 = 1 λ 2 size 12{λ rSub { size 8{1} } = { {1} over {λ rSub { size 8{2} } } } } {} and the form of the homogeneous solution is

v o [ n ] = A 1 λ 1 n + A 2 λ 1 n size 12{v rSub { size 8{o} } \[ n \] =A rSub { size 8{1} } λ rSub { size 8{1} rSup { size 8{n} } } +A rSub { size 8{2} } λ rSub { size 8{1} rSup { size 8{ - n} } } } {}

The two terms of the homogeneous solution are shown below.

Thus, the homogeneous solution consists of a linear combination of decaying and growing geometric functions whose rates of decrease and increase are the same. Increasing the quantity rg increases the magnitude of the rate of change of voltage.

2/ Particular solution

Next we find a particular solution to the difference equation

k = 0 K a k y p [ n + k ] = l = 0 L b l x [ n + 1 ] size 12{ Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } y rSub { size 8{p} } \[ n+k \] } = Sum cSub { size 8{l=0} } cSup { size 8{L} } {b rSub { size 8{l} } x \[ n+1 \]} } {}

for n>0 and

x [ n ] = Xz n size 12{x \[ n \] = ital "Xz" rSup { size 8{n} } } {}

where z does not equal one of the natural frequencies. We assume that

y p [ n ] = Yz n size 12{y rSub { size 8{p} } \[ n \] = ital "Yz" rSup { size 8{n} } } {}

and we solve for Y . Substitution for both x[n] and y[n]in the difference equation yields, after factoring,

( k = 0 K a k z k ) Yz n = ( l = 0 L b l x l ) Xz n size 12{ \( Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } z rSup { size 8{k} } \) ital "Yz" rSup { size 8{n} } = \( Sum cSub { size 8{l=0} } cSup { size 8{L} } {b rSub { size 8{l} } x rSup { size 8{l} } } \) ital "Xz" rSup { size 8{n} } } } {}

After dividing both sides of the equation by z n size 12{z rSup { size 8{n} } } {} we can solve for Y which has the form

Y = H ˜ ( z ) X size 12{Y= { tilde {H}} \( z \) X} {}

where

H ˜ ( z ) = l = 0 L b l z l k = 0 K a k z k size 12{ { tilde {H}} \( z \) = { { Sum cSub { size 8{l=0} } cSup { size 8{L} } {b rSub { size 8{l} } z rSup { size 8{l} } } } over { Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } z rSup { size 8{k} } } } } } {}

3/ System function

H ˜ ( z ) = l = 0 L b l z l k = 0 K a k z k zeros poles size 12{ { tilde {H}} \( z \) = { { Sum cSub { size 8{l=0} } cSup { size 8{L} } {b rSub { size 8{l} } z rSup { size 8{l} } } } over { Sum cSub { size 8{k=0} } cSup { size 8{K} } {a rSub { size 8{k} } z rSup { size 8{k} } } } } `` { { dlarrow `` ital "zeros"} over { dlarrow `` ital "poles"} } } {}

  • is called the system function
  • is a rational function in z that has poles and zeros
  • has poles that are the natural frequencies of the system
  • is a skeleton of the difference equation
  • characterizes the relation between x[n] and y[n]

a/ Example—reconstruction of difference equation from H ˜ ( z ) size 12{ { tilde {H}} \( z \) } {}

Suppose

H ˜ ( z ) = Y X = z z + 1 size 12{ { tilde {H}} \( z \) = { {Y} over {X} } = { {z} over {z+1} } } {}

what is the difference equation that relates y[n] to x[n]? Crossmultiply the equation and multiply both sides by z n size 12{z rSup { size 8{n} } } {} to obtain

( z + 1 ) Y z n = z X z n size 12{ \( z+1 \) Y`z rSup { size 8{n} } =z`X`z rSup { size 8{n} } } {}

which yields

( z n + 1 Y + z n Y = z n + 1 X size 12{ \( z rSup { size 8{n+1} } Y`+z rSup { size 8{n} } Y=z rSup { size 8{n+1} } `X} {}

from which we can obtain the difference equation

y[n+1] + y[n]= x[n+1]

b/ Pole-zero diagram

H ~ size 12{ {H} cSup { size 8{ "~" } } } {} characterizes the difference equation and H ~ size 12{ {H} cSup { size 8{ "~" } } } {} is characterized by K + L + 1 numbers: K poles, L zeros, and one gain constant. Except for the gain constant, H ~ size 12{ {H} cSup { size 8{ "~" } } } {} is characterized by a pole-zero diagram which is a plot of the locations of poles and zeros in the complex-z plane.

4/ Total solution

The general solution is

y [ n ] = k = 1 K A k λ k n + X H ˜ ( z ) z n for n > 0 size 12{y \[ n \] = Sum cSub { size 8{k=1} } cSup { size 8{K} } {A rSub { size 8{k} } λ rSub { size 8{k} rSup { size 8{n} } } } +X { tilde {H}} \( z \) z rSup { size 8{n} } `` ital "for"``n>0} {}

and

y[n] = 0 for n<0.

The general solution can be written compactly as follows

y [ n ] = ( k = 1 K A k λ k n + X H ˜ ( z ) z n ) u [ n ] size 12{y \[ n \] = \( Sum cSub { size 8{k=1} } cSup { size 8{K} } {A rSub { size 8{k} } λ rSub { size 8{k} rSup { size 8{n} } } } +X { tilde {H}} \( z \) z rSup { size 8{n} } \) u \[ n \]} {}

5/ Initial conditions

To completely determine the total solution we need to determine the K coefficients { A 1 , A 2 , . . . A K size 12{A rSub { size 8{1} } ,A rSub { size 8{2} } , "." "." "." A rSub { size 8{K} } } {} }. These are determined from K initial conditions which must be specified. These conditions result in a set of K algebraic equations that need to be solved to obtain the initial conditions so that the total solution can be specified. We shall find another, and simpler, method to determine the total solution later.

Example — discretized CT system

We have previously considered the discretized approximation to a lowpass filter.

The equilibrium equation is

dv o ( t ) dt = 1 RC v o ( t ) + 1 RC v i ( t ) size 12{ { { ital "dv" rSub { size 8{o} } \( t \) } over { ital "dt"} } = - { {1} over { ital "RC"} } v rSub { size 8{o} } \( t \) + { {1} over { ital "RC"} } v rSub { size 8{i} } \( t \) } {}

We know that the unit step response of this network starting from initial rest is

v o ( t ) = ( 1 e t / RC ) u ( t ) size 12{v rSub { size 8{o} } \( t \) = \( 1 - e rSup { size 8{ - t/ ital "RC"} } \) u \( t \) } {}

We showed that a discretized approximation to this system yields the difference equation

v o [ n + 1 ] = ( 1 β ) v o + αv i [ n ] size 12{v rSub { size 8{o} } \[ n+1 \] = \( 1 - β \) v rSub { size 8{o} } +αv rSub { size 8{i} } \[ n \]} {}

where α = T/(RC). We will determine the solution by finding the homogeneous and particular solution. But all the information we need is contained in the system function which we can obtain by substituting v i [ n ] = V i z n size 12{v rSub { size 8{i} } \[ n \] =V rSub { size 8{i} } z rSup { size 8{n} } } {} and v o [ n ] = V o z n size 12{v rSub { size 8{o} } \[ n \] =V rSub { size 8{o} } z rSup { size 8{n} } } {} into the difference equation to obtain

so that

H ˜ ( z ) = V ˜ o ( z ) V ˜ i ( z ) = α z ( 1 α ) size 12{ { tilde {H}} \( z \) = { { { tilde {V}} rSub { size 8{o} } \( z \) } over { { tilde {V}} rSub { size 8{i} } \( z \) } } = { {α} over {z - \( 1 - α \) } } } {}

The natural frequency is λ = 1 α size 12{λ=1 - α} {} so that the solution has the form

v o [ n ] = ( A ( 1 α ) n + H ˜ ( 1 ) ( 1 ) n ) u [ n ] size 12{v rSub { size 8{o} } \[ n \] = \( A \( 1 - α \) rSup { size 8{n} } + { tilde {H}} \( 1 \) \( 1 \) rSup { size 8{n} } \) u \[ n \] } {}

where we have made use of the fact that u [ n ] = 1 n u [ n ] size 12{u \[ n \] =" 1" rSup { size 8{n} } u \[ n \]} {} . Since H ~ size 12{ {H} cSup { size 8{ "~" } } } {} (1) = 1,

v o [ n ] = ( A ( 1 α ) n + 1 ) u [ n ] size 12{v rSub { size 8{o} } \[ n \] = \( A \( 1 - α \) rSup { size 8{n} } +1 \) u \[ n \]} {}

Finally, the initial condition, v o [ 0 ] = 0 size 12{v rSub { size 8{o} } \[ 0 \] =" 0"} {} , implies that A = −1 so that the solution is

v o [ n ] = ( 1 1 α ) n ) u [ n ] size 12{v rSub { size 8{o} } \[ n \] = \( 1 - 1 - α \) rSup { size 8{n} } \) u \[ n \]} {}

which is the same result we obtained earlier by solving the difference equation iteratively.

We compare the step response of the CT system with the DT approximation

v o ( t ) = ( 1 e t / RC ) u ( t ) size 12{v rSub { size 8{o} } \( t \) = \( 1 - e rSup { size 8{ - t/ ital "RC"} } \) u \( t \) } {} and v o [ n ] = ( 1 ( 1 T RC ) n ) u ( t ) size 12{v rSub { size 8{o} } \[ n \] = \( 1 - \( 1 - { {T} over { ital "RC"} } \) rSup { size 8{n} } \) u \( t \) } {}

The solutions are shown below for RC = 1 and T/(RC) = 0.1.

XI. CONCLUSIONS

  • Systems are typically described by an arrangement of subsystems each of which is defined by a functional relation. Systems are classified according to such properties as: memory, causality, stability, linearity, and time-invariance. Linear, time-invariant systems (LTI) are special systems for which powerful mathematical methods of description are available.

Logic for an analysis method for LTI systems

  • H(s) characterizes system size 12{ drarrow } {} compute H(s) efficiently.
  • In steady state, response to Xe st size 12{"Xe" rSup { size 8{"st"} } } {} is H ( s ) Xe st size 12{H \( s \) "Xe" rSup { size 8{"st"} } } {}
  • Represent arbitrary x(t) as superpositions of Xe st size 12{"Xe" rSup { size 8{"st"} } } {} on s.
  • Compute response y(t) by superposing H ( s ) Xe st size 12{H \( s \) "Xe" rSup { size 8{"st "} } } {} on s.

1/ Structural versus functional descriptions

Just as with CT systems, DT systems can be described either structurally, with a block diagram or a network diagram, or functionally by a system function.

H ~ ( z ) size 12{ {H} cSup { size 8{ "~" } } \( z \) } {} characterizes system

2/ Steady-state response to zn is particularly simple

Since the steady-state response to a complex geometric (exponential) is so simple, it is desirable to represent arbitrary DT functions as sums (integrals) of building-block complex geometric (exponential) functions chosen so that steady-state dominates. The steady-state response to each complex geometric (exponential) is readily computed. For a DT LTI system, the response to an arbitrary input can be computed by superposition.

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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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