Question

All the questions are related to linear system theory.(Book: Signals and systems 2nd edition)

Q1. What is difference between Fourier Series and Fourier Transform?

Q2. What is difference between Fourier transform, Laplace transform and Z transform?

Q3. Why we are interested in exponential signals?

Q4. How to understand causal system? If a LTI system is non-causal, what does it mean in real life?

Q5. What is the difference of the LTI system output when the input x(t)= exp(-2t) is changed to x(t)=exp(-2t)u(t)?

Q6. How to understand Region of convergence?

Q7. Why we need transforms?

Q8. After a continuous signal is converted to the discrete signal after sampling, what is the impact on the transform domain accordingly?

Answer 1

ANSWER FOR Q1:

Fourier series and Fourier transform are both given by Fourier.The main difference between them is Fourier series is applied to Periodic signals and Fourier transform was applied to Non-Periodic Signals.

ANSWER FOR Q2:

1)For any continuous signal x(t),Fourier transform can be given as

For any discrete signal x[n],Fourier transform can be given a   

     

where N is the length of the signal x[n].

2)For any signal x(t),laplace transform is given as follows

the laplace transform applicable for continuous signals only.

3)The Z-transfrom of any discrete signal x[n] is given as

• The laplace transform was applicable for continuous signals only.
• The fourier transform applicable for both continuous and discrete signals.
• The Z-transform applicable for only discrete signals

These are the main differences between Fourier,Laplace,and Z-transforms

ANSWER FOR Q3:

An exponential signal can be expressed in terms of sine and cosine functions which are important in making easy calculations where other signals like unit signal,ramp signal,square signal are not as efficient as sine and cosine signals.Calculation with exponential signal will also easier than other signals mentioned above.Thats why an exponential signal is preferred.

ANSWER FOR Q4:

A system is said to be Causal if its output depends on present inputs and past inputs but not on future inputs.These systems were practically realizable,in real time applications only present and past inputs or samples present. EXAMPLES:

1.y[n]=x[n] is a causal system  whose output depends on present inputs

2.y[n]=x[n-1] is also a causal system whose output depends on past inputs

If an LTI system is said to be non-causal then its present output depends on future inputs.That means in real life in practical cases it is not possible to implement on these non-causal systems.

EXAMPLES:

1.y[n]=x[n+1] is non-causal LTI sytem whose output depends on future inputs