Signals and Systems

(DEF) Signal : A signal is formally defined as a function of one or more variables, which conveys information on the nature of physical phenomenon.

Continuous and discrete-time signalsEven and odd signalsPeriodic signals, non-periodic signalsDeterministic signals, random signalsCausal and anticausal signalsRight-handed and left-handed signalsFinite and infinite length

Continuous and discrete-time signalsContinuous signal - It is defined for all time t : x(t)Discrete-time signal - It is defined only at discrete instants of time : x[n]=x(nT)

Even and odd signalsEven signals : x(-t)=x(t)Odd signals : x(-t)=-x(t)Even and odd signal decomposition xe(t)= 1/2(x(t)+x(-t)) xo(t)= 1/2(x(t)-x(-t))

Periodic signals, non-periodic signalsPeriodic signals - A function that satisfies the condition x(t)=x(t+T) for all t - Fundamental frequency : f=1/T - Angular frequency : = 2/T

Non-periodic signals

Deterministic signals, random signalsDeterministic signals -There is no uncertainty with respect to its value at any time. (ex) sin(3t)

Random signals - There is uncertainty before its actual occurrence.

Causal and anticausal SignalsCausal signals : zero for all negative timeAnticausal signals : zero for all positive timeNoncausal : nozero values in both positive and negative timecausal signalanticausal signalnoncausal signal

Right-handed and left handed-signal : zero between a given variable and positive or negative infinity

Finite-length signal : nonzero over a finite interval tmin< t< tmaxInfinite-length singal : nonzero over all real numbers

(DEF) System : A system is formally defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.

Communication systemControl systemsRemote sensing systemBiomedical system(biomedical signal processing)Auditory system

Communication system

Control systems

Papero

Remote sensing systemPerspectival view of Mount Shasta (California), derived from a pair of stereo radar images acquired from orbit with the shuttle Imaging Radar (SIR-B). (Courtesy of Jet Propulsion Laboratory.)

Biomedical system(biomedical signal processing)

Auditory system

Discrete-time signals:

Time: defined only at discrete values of time:

Descriptions: sequences of real or complex numbers , Note A.: they take on values in the continuous interval , Note B.: sampling of analogue signals:sampling interval, period: , sampling rate: number of samples per second,sampling frequency (Hz): .

*Functional representation:

C. Tabular representation:

n-2-1012x(n)0.122.011.785.230.12

Elementary Discrete-Time SignalsA. Unit sample sequence (unit sample, unit impulse, unit impulse signal)

B. Unit step signal (unit step, Heaviside step sequence)

C. Complex-valued exponential signalwhereandT is sampling period and is sampling frequency.(complex sinusoidal sequence, complex phasor)

d. Sinusoidal sequence

e. Real-valued exponential sequence

f. Complex-valued exponential sequence

Sequence energy Sequence power

The periodicity of sequence

The periodicity of sinusoidal sequence

Any arbitrary real-valued sequence can be decomposed into its even and odd component: Even and odd synthesis

Basic Operations on SignalsShiftingTime reversalTime scalingScalar multiplicationSignal multiplierSignal addition

Operations on sequence Time-shifting operation

Time-reversal (folding) operation Addition operation

Scaling operation Product (modulation) operation

Sample summation Sample production

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