EEEN 214 Signals and Systems Lecture Notes

Flashcards for EEEN 214 Signals and Systems Lecture Notes

Continuous-time exponential signal

Can be a constant, a monotonic exponential, exponentially varying sinusoids, or sinusoids (complex frequency).

Fundamental Period (T0)

The smallest value of T for which x(t + T) = x(t) for all t.

Harmonically related complex exponentials

Represented as φk(t) = e^(jkω0t), where k is an integer.

Discrete-time exponential signal

Signal of the form x[n] = α^n, where α can be real or complex.

Periodicity of discrete-time exponential signal

Signals with frequencies ω0 and ω0 + k2π are identical.

Condition for periodicity in discrete-time signals

For a signal to be periodic, ω0 / (2π) must be a rational number.

Fundamental period of discrete-time signal

If N and m have no common factors, then N = (2π) / ω0, where N is the fundamental period.

Memoryless System

Output at a given time only depends on the input at that same time.

Invertible System

Distinct inputs lead to distinct outputs.

Causal System

At any time, the output depends on current and past (but not future) inputs.

BIBO Stability

Bounded input results in a bounded output.

Time Invariant System

System behavior is fixed over time.

Linear System

Satisfies the superposition property.

Energy Signal

A signal with finite energy (E < ∞).

Power Signal

A signal with finite power (0 < P < ∞).

Linear Time-Invariant (LTI) Systems

Systems that possess the superposition property and are time invariant.

Unit-impulse function in LTI systems

Plays a fundamental role because very general signals can be represented as linear combinations of delayed impulses.

Sifting property of the unit impulse

The unit impulse δ[n] samples the value of a signal at n = 0.

Convolution

The process by which the output of an LTI system is obtained by combining the input signal with the system impulse response.

Impulse Response h[n]

The response of a system to the unit impulse δ[n].

Commutativity of LTI systems

x * h = h * x

Distributivity of LTI systems

x * (h1 + h2) = x * h1 + x * h2

Associativity of LTI systems

x * (h1 * h2) = (x * h1) * h2 = x * h1 * h2

BIBO Stability Condition (Continuous Time)

The impulse response h(t) is absolutely integrable: ∫|h(t)| dt < ∞

BIBO Stability Condition (Discrete Time)

The impulse response h[n] is absolutely summable: Σ|h[n]| < ∞

Causality Condition (Continuous Time)

h(t) = 0, for t < 0

Causality Condition (Discrete Time)

h[n] = 0, for n < 0

Memorylessness Condition (Continuous Time)

h(t) = Aδ(t)

Memorylessness Condition (Discrete Time)

h[n] = Aδ[n]

Unit-step response

The response of an LTI system when the the unit step function is used as an input.