Homework - 5

Today's homework is more like recollecting a few important points from signals and systems. In case you do not understand a point, you are advised to go back to the text books or knowledgeable friends. In case you have a problem, relax and come to the next class with an open mind.

Good Day

1. Consider the analog signal x(t) = 3cos(100.pi.t). Determine the minimum sampling rate that would restore the signal normally.
Remember and revise :

• Signals and systems are often more usefully described by their frequency-domain properties than their time-domain properties.
• The more a signal is localised in one domain (time or frequency), the less it is localised in the other domain.
• Convolution and multiplication of functions are dual operations in the time and frequency domain.
• The Fourier transform of a periodic signal consists only of impulses.
• Signal energy is conserved in the Fourier transformation process (Parseval's theorem).
• The Fourier series expresses a periodic signal as a sum of sinusoids at harmonics of the fundamental frequency of the signal.
• The sinusoids used in the Fourier series to represent a signal are all orthogonal to each other.
• If a stable Linear Time-Invariant (LTI) system is excited by a periodic signal, the response  is also periodic signal with the same fundamental frequency.
• If a signal is discrete in one domain, it is periodic in another.
• A shift in one domain corresponds to a multiplication by a complex exponential in another domain.
• If a continuous-time signal is sampled to form a discrete-time signal , the Discrete-time Fourier Transform (DTFT)(will be covered next week) of the dicrete-time signal can be found from the Continuous-time Fourier transform(CTFT) of the continuous-time signal by a change of variable, but the converse is not generally true.
• Every LTI system is completely characterised by its impulse response.
• The response of an LTI system to an arbitrary (any) signal can be found by 'convolving' the signal with the system impulse response.
• The impulse response of a cascade connection of LTI systems is the convolution of the individual impulse responses.
• The impulse response of a parallel connection of LTI systems is the sum of the individual impulse responses.
• A LTI system is BIBO(Bounded input produces bounded output) stable if its impulse response is absolutely summable.
• LTI systems can be represented by block diagrams and this type of representation is useful both in synthesising systems and in understanding their dynamic-behaviour.

Bye