Signals	Systems	System Response
Frequency Representation	Fourier Series / Transforms	Background Material

What is a System?

A system is a process that may influence a signals. We represent systems in graphical form by input-output relationships.

For convenience, input signals are often labeled "x" while output signals are labeled "y." This is definitely a recommendation rather than a rule. We will endeavor to use symbols to promote clarity.

Bank Account Interest

A bank account that yields interest on money deposited. To model this system, let us consider the input to be the amount of money deposited . The output is the balance in the bank account (let's neglect withdrawls).

For example, if $1000 was deposited in a bank account at time=0 with 5% interest. Suppose the interest is paid in discrete cycles (such as annually). Then, an equation governing the output of the investment is:

We can graph the output using a discrete time plot:

In this course, we will learn how to generalize this type model for variable rates of deposits, withdrawls, and interest rates.

For a system is linear, it must exhibit addivity and scaling (homogeneity).

Additivity means that when the sum of different inputs are given to a system, the result is the sum of the outputs from each input separately:

Scaling (homogeneity) means that when one multiplies a input to a system by a scalar, the output is also multipled by the same scalar:

Time Invariance

A system that is time invariant has the property that a time shift in the given input will produce the same time shift in the output. Let’s denote an arbitrary time shift by the constant t0:

http://en.wikipedia.org/wiki/Time-invariant_system

BIBO Stability

In this course, we stress the concept of BIBO stability. This means that for every bounded input the system produces a bounded output. In BIBO stability, unbounded inputs may result in unbounded output signals.

Ex1. Is BIBO stable? Yes. |y(t)| can never be larger than |x(t)|, and if x(t) is bounded, then y(t) must be a finite.

Ex2. Is BIBO stable? No. If |x(t)|<M, then |y(t)|>|1/M|, so y(t) can be arbitrarily large even when x(t) is bounded.

Causality

A system is causal if the output depends only on the input at the present or past time, but not in the future.

Invertibility

A system is invertible if the input to the system can be recovered from the output. This implies the existence of an inverse system that takes the output of the original system as its input and produces the input of the original system.

For ex.

Thus, for the continuous case,

or for the discrete case,

Linear and Time Invariant Systems: A VERY Special Case

A few systems fall under a special category that makes predicting the output VERY easy. These systems have two essential properties: Linearity and Time Invariance and they are known as LTI Systems. For a system to be LTI, you must individually show additivity, scaling, and time invariance.