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Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE) PDF Download

Continuous Rational System 

Causality


For a causal LTI system, the impulse response is zero for t=0 (and thus it is right sided!) h(t) = 0 for all t<0

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)
where H(s) is the system function (assuming system has a system function
 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

Thus, if the region of convergence is non null,  must be included in the ROC for the system to be causal. 

 Proof: As region of convergence is not null there exist an Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)  

Hence H(S) is convergent for all Re(S)>Re(So)

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)  is inthe region and belongs to ROC

Necessary and sufficient condition for causality in a rational system: The region of convergence must include 

The region of convergence must includeAnalysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 

DISCRETE RATIONAL SYSTEM :


A Causal Discrete time LSI system has an impulse response h[n], this is zero for n<0 and thus it is right sided.h[n]=0 for all


n< 0 for causality system function H(z) Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 Assuming H(z) has a non null ROC ,we require,

  Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)  for n > 0,

thus  must be included in the ROC.

 Example:

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)  

STABILITY OF RATIONAL SYSTEMS:


A continous LSI system is stable if and only if its impulse response is absolutely integrable, i.e.

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)
.
Exploring the convergence of Laplace transform of impulse response of a stable LSI system, we find that

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 

hus H(s) converges on imaginary axis ( Re(s)=0 ) So, Re(s)=0 or imaginary axis is contained in ROC of system function for all stable LSI systems.


We can also look at this from a different point of view.Impulse response being absolutely integrable implies Fourier transform converges as  is nothing but Fourier transform it is also bound to converge for Re{s} = 0 Re{s} = 0 is included in its ROC.


In general, Re{s} = 0 lies in ROC is not sufficent condition to imply stability. But for rational systems Re{s} = 0 lies in ROC system is stable.
Now, we will prove the above result .


Proof for sufficiency condition :-

For any system to be stable, poles can not lie in ROC.Thus, there should not be any poles on the (imaginary axis) Re(s)=0.


Suppose α and β are the poles of the system function H(s) where Re(α)<0 and Re( β)>0.
Now consider, inverse transform of , Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE) there are two choices

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 As Re{s}=0 is contained in the ROC and , the only possible option is  ( to have a non-empty ROC).

Looking at inverse transform of  Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 

As Re{s} = 0 lies in ROC , we will have to take to be the inverse.


Thus, in a rational system, with ROC of the system function including Re(s)=0, the poles to the left of imaginary axis contribute rightsided exponentially decaying term and poles to the right of the imaginary axis contribute left-sided exponentially decaying term.

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)
α contributes a right-sided decaying exponential

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)
β contributes a left handed decaying exponent.

 

Poles to the right of imaginary axis contribute -Pβ(t)eβtu(-t), where Pβ(t) is a polynomial of degree k-1 k = order of pole at  in H(s) Similarly poles to the left of imaginary axis

contribute Pα(t)eαtu(t)


The absolute integral Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 

Thus the absolute integral  Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)  sum of the absolute integrals of these terms (finite number because the system function is rational) <∞

Therefore, the system is stable.
Later, we shall prove the theorem , that irrespective of the polynomialAnalysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)converges  if and only if Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE) converges, in order to justify the convergence of each absolute integral.

 Rational continuous system functions 

Let the system function

 Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

We can represent graphically the system function by showing all poles(zero's of D(s) ) and zero's (zero's of H(s) ) and it's ROC in splane. eg: 

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

Complete pictorial representation of the above system function H(s) in s-plane.

 Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

From this graph we can write the system function as 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

Now for stability, Re{s} = 0 should lie in ROC.

Representation of poles and zeros 

Consider representation of the system function

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 

Thus H(s) can be represented as . Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 


On expansion of H(s) in terms of partial fraction we would get

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 

Recall that in a rational system, with ROC of the system function including Re(s)=0, the poles to the left of imaginary axis contribute right-sided exponentially decaying term and poles to the right of the imaginary axis contribute left-sided exponentially decaying term.


Thus, as we have seen earlier, α contributes a right handed decaying exponential and β contributes a left handed decaying exponential and the contributions of following terms in the denominator are

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 

Theorem


Irrespective of the polynomial Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)  converges if and only if Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)  converges.

 

 

Proof by induction:

Mathematical Induction on degree of polynomial

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 

Base case: Suppose the statement is true for n=1 case we prove it is true for n=2 case.

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

Induction step: We assume  Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE) converges , for any polynomial of degree (k-1), We proceed to proveAnalysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE) converges for
apolynomial of degree k

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE) is polynomial of degree (k-1) , by the assumption , we knowAnalysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE) converges there by 
Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)also converges.

Hence, proved.

Theorem 2

For a discrete rational system stability implies and is implied by the unit circle in the z plane belonging to the ROC of the system function.

Proof :-

(a) For the stability of the system function If the discrete rational system is stable then

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

The z transform of the impulse response (or the system function ) converges for | z | = 1.

(b) For a stability to be implied by | z | =1 (the unit circle ) belonging to the ROC of the system function

A pole cannot lie on the unit circle | z | = 1 in a stable system.

 Rational discrete system functions.
Considering the function

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 when is the pole of order 1 { ( α < 1) is the assumption } ,  β is a pole of order 1 (β >1) .

Now consideringthe Inverse transform of Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)  we have,

 Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

as | z | =1 is contained in the ROC and <1, hence the only possible option for inverse is .
Similarly for the function

 Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)


( since β >1and | z | =1 is contained in the ROC of the function )
Therefore,

The contribution of  is a right sided exponentially decaying term (possibly multiplied by a polynomial in n if the order of pole >1 ) The contribution of  is a left sided exponentially decaying term ( possibly multipled by a polynomial in n if the order of the pole >1 )

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)    Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 

Proof for stability of rational discrete systems Similar to proof for stability of rational continuous systems, the absolute sum must be convergent.
The absolute sum Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

(Assuming two poles  α( α<1) and  β( β>1) of the order >1)

 

Increasing the number of poles would not make any difference to the proof .pα and  p β  are polynomials in n.


Now we know that Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)   are absolutely summable. Finite number of such terms is absolutely summable and hence the Impulse response is absolutely summable.

Therefore ,the system is stable.

The absolute summability of one sided terms of   Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE) (where p(n) is a polynomial) depends only onAnalysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)  and not on the polynomial.

 

 Theorem 4

We prove summability of depends on summability of .

Proof by Induction: Induction on degree of polynomial

Base case:  (k=1) Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

Induction step: Assume Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)  is summable for (k-1) case .we proceed to prove it for k case.

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

by our assumption is summable for polynomial of degree (k-1).

 

THEOREM :


A neccesary and sufficient conditon for a continuos rational system to be a Causal and Stable is that all the poles must lie in the left half plane, i.e. Re (s)< 0.


THEOREM :


A neccesary and sufficient conditon for a discrete rational system to be a Causal and Stable is that all the poles must lie inside the unit circle, i.e.|z| < 1 .
System Defination of Causal Rational System and Linear Constant Coefficient Difference equation

(a) Continuos system :-The system function can be written as ,

 

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 Taking the inverse Laplace transform we have ,

 Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

(b) Discrete system :-

Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

 It is always possible to write the system function this way for a Causal rational discrete system .

 Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)
Taking the inverse z-transform of the above equation we have ,

 

Conclusion:

In this lecture you have learnt:

 Necessary and sufficient condition for causality in a continuous rational system : The region of convergence must include Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)

  • Necessary and sufficient condition for causality in a discrete rational system: The region of convergence must include Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE)
  • In general, Re{s} = 0 lies in ROC is not sufficent condition to imply stability. But for rational systems Re{s} = 0 lies in ROC system is stable.
  • In a rational system, with ROC of the system function including Re(s)=0, the poles to the left of imaginary axis contribute right-sided exponentially decaying term and poles to the right of the imaginary axis contribute left-sided exponentially decaying term.
  • For a discrete rational system stability implies and is implied by the unit circle in the z plane belonging to the ROC of the system function.
The document Analysis of LTI Systems with Rational System Functions | Signals and Systems - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Signals and Systems.
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FAQs on Analysis of LTI Systems with Rational System Functions - Signals and Systems - Electrical Engineering (EE)

1. What is an LTI system?
An LTI (Linear Time-Invariant) system is a type of system in signal processing and control theory that possesses two key properties: linearity and time-invariance. Linearity means that the response of the system to a linear combination of inputs is equal to the linear combination of the responses to each individual input. Time-invariance means that the system's response remains the same regardless of when the input is applied.
2. How can LTI systems be represented mathematically?
LTI systems can be represented mathematically using rational system functions, which are ratios of two polynomials. The numerator polynomial represents the output of the system, while the denominator polynomial represents the input. The coefficients of these polynomials determine the behavior of the system.
3. What is the importance of analyzing LTI systems with rational system functions?
Analyzing LTI systems with rational system functions allows us to understand their behavior, stability, and response to different inputs. By studying the properties of rational system functions, we can determine the system's frequency response, impulse response, and overall transfer function. This analysis is crucial in various fields, including telecommunications, control systems, and signal processing.
4. How can we determine the stability of an LTI system with a rational system function?
The stability of an LTI system can be determined by examining the poles of its rational system function. If all the poles have negative real parts, the system is stable. Conversely, if any pole has a positive real part, the system is unstable. The location of the poles also provides insights into the system's transient and steady-state response.
5. What are some practical applications of analyzing LTI systems with rational system functions?
Analyzing LTI systems with rational system functions is essential in various engineering applications. It helps in designing filters for signal processing, understanding the behavior of control systems, modeling communication channels, and designing equalizers for audio systems. Additionally, it aids in understanding the stability and response characteristics of systems in fields such as electronics, aerospace, and robotics.
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