Digital Signal Processing

What is a signal?

• A signal is a carrier of information, any sort of physical or computable quantity upon which we bestow a meaning.

• A digital signal is a signal whose values are discrete numbers. This is direct opposite to continuous time or analog signals, which have value at any point in time.

Digitals Signals

• Due to their representation by numbers within a computer ’s memory can only have a finite number of values for the amplitude.

• The also only exists in certain instance of time (or space)

• This is why we can represent them as sequences or discrete functions.

System and Signals

A system is anything can modify a signal. This can be a physical system, a digital filter and often is represented by a function or a transform operator:

Y=F(x)

Linear, Shift- Invariant (LSI) Systems:

• Linear-superposition applies:

If F(x) =g and F(y)= h

• Shift Invariant –shape of system output does not vary with a start point:

If x(n)->y(n), then x(n-n0) -> y(n-n0)

Sequences Properties

• Periodicity

A sequence is periodic if there exists a period T such that, for all n and k,

x(n+kT) =x(n)

• Symmetry

A symmetry is even if for every n, x(n)=x(-n)

A symmetry is odd if for every n,x(n)= - x(-n)

A symmetry is conjugate symmetric if, for all n, x(n)=x*(-n)

A symmetry is conjugate anti symmetric if, for all n, x(n)=-x*(-n)

Fundamental Sequences

• Unit sample (digital equivalent of the Dirac Delta):

• δ(n) = 1 when n=0

• δ(n) = 0 otherwise

• Unit step (digital equivalent of the Heaviside Step Function)

u(n) = 1 when n>=0

u(n) = 0 otherwise

miscellaneous definitions

Stability: A system is said to be stable if the outp

ut is bounded for all bounded inputs.

Causality: A system is causal if its current output depends only on the current input, previous input and/or previous outputs.