﻿ ESE 305

# ESE 305 : Deterministic Signals and Systems

### Section 1.1 : Introduction

Definition Model
Signals Are transformed by Systems into other Signals. Can be modeled as Column Vectors.
Systems Transform Signals into other Signals. Can be modeled as Matrices or Operators.

### Section 1.2 : CT, DT, & Digital

Signals Definition Model
Continuous-Time (CT) Domain and range is continuous. Functions OR Real Valued Continuous Column Vectors.
Discrete-Time (DT) Domain is discrete, but range is continuous. Real Valued Column Vectors.
Digital Signals Domain and range is discrete. Rational Valued Column Vectors.

Output
Discrete Continuous
Input Discrete Digital DT
Continuous CT

### Section 1.3 : CT

• : Functions
• Positive-time : x(t)=0 for t<0
• Negative-time : x(t)=0 for t>0
• q(t) : Unit Step Function
• Magnitude : + or 0
• Amplitude : + or 0 or –

### Section 1.4 : CT Manipulation

• Manipulation : x(t)'s argument may be a function of t
• Manipulation : Select different t's and substitute

### Section 1.4.1 : Shifting & Flipping

• Shifting :
• x(t-1) : Shifts x(t) to the right by 1.
• x(t+1) : Shifts x(t) to the left by 1.
• Flipping :
• x(-t) : Flips x(t) over the t axis.
• Shifting & Flipping :
• x(-t-1) : The flip of x(t+1).
• x(-t+1) : The flip of x(t-1).

### Section 1.4.2 : Multiplication & Addition

• x1(t) * x2(t) : is a new whose value at each instant is the product of x1 and x2 at that instant
• x1(t) + x2(t) : is a new whose value at each instant is the sum of x1 and x2 at that instant

### Section 1.4.3 : Modulation

• Modulation : Allows multiple to be sent simultaneously over the same line
• x(t) : Modulating
• ωc : Carrier Frequency
• cos(ωct) : Carrier
• x(t) cos(ωct) : Modulated

### Section 1.4.4 : Windows and Pulses

• Rectangular Window : A that is constant from -a to a, and zero otherwise
• Shifted Window : A Rectangular Window shifted by "a"
• Pulse : A Window with unit area

### Section 1.5 : Impulse

• Impulse : A zero-width
• δ(t) : Impulse
• δ(t) : (d/dt)
• x(t0) : ∫x(t)δ(t-t0)dt (Sifting Property)

### Section 2.1 : Intro to Systems

• System : A black box with 1 input and 1 output.

### Section 2.2 : CT & Memory

• Input : Signal u(t)
• Output : Signal y(t)
•  Type y(t₀) depends on Memoryless Causal Noncausal u(t< t0) No Maybe Maybe u(t=t0) Maybe Maybe Maybe u(t>t0) No No Yes

### Section 2.3 : State - Initial Conditions

• Memory : A system s.t. y(t) depends on knowing past input (or future)
• State : A summary of a system's past input u(t<0) into State Variables [S.V.] (Initial Conditions) x(t0)
• y(t) depends on ...  Causal : u(t≤0) Non-Causal : u(t) Memoryless : u(0) u(0) AND u(t≥0) AND u(t<0) OR Distributed System : Infinite number of S.V. Lumped System : Finite number of S.V.

### Section 2.3.1 : Zero-Input Response & Zero-State Response

• Initially Relaxed : A system whose x(t0) = 0
• Zero-State Response : The response y(t) of an Initially Relaxed System
• Zero-State Response : A system's response y(t) generated when x(t0) = 0 , AKA Forced Responce
• Zero-Input Response : A system's response y(t) generated when u( t ) = 0 , AKA Unforced Responce

### Section 2.4 : Linearity of Memoryless Systems

• A Memoryless System is Linear iff (1 & 2) or 3
1. u1(t) + u2(t) → y1(t) + y2(t) (Additivity)
2. αu1(t) → αy1(t) (Homogeneity)
3. α1u1(t) + α2u2(t) → α1y1(t) + α2y2(t) (Superposition)
• Linear and Time-Varying is possible with α(t).
• Linear Systems : can be represented as y(t) = Matrix * u(t)

### Section 2.4.1 : Linearity of Systems with Memory

• If a System is Linear:
•  x1(t) + x2(t)} u1(t) + u2(t)}
•  αx1(t)} αu1(t)}
→ αy1(t) (Homogeneity)
• Total Responce: Zero-input responce + Zero-state responce

### Section 2.5 : Time Invariance and its Implications

• Time Shifting Property : Output(Shift(Input)) = Shift(Output(Input))
• Time Invariant System : A System that has the Time Shifting Property

### Section 2.6 : Implications of Linearity and Time Invariance - Zero-State Responces

• Memoryless LTI : y=au
• Given u1 and y1 : We can reconstruct "a" and therefore the System's Function
• LTI : y[n] = ... + a-2u[n-2] + a-1u[n-1] + a0u[n] + a+1u[n+1] + a+2u[n+2] + ...
• Given u1 and y1 : We can reconstruct all "ai" and therefore the System's Function

### Section 3.1 : Intro to Convolutions, Difference & Differential EQNs

• Total Responce: Zero-state responce + Zero-input responce

### Section 3.1.1 : Preliminaries

• Memoryless LTI : y=au
• Causal LTI : y[n] = au[n] + bu[n-1] + cu[n-2] + ...

### Section 3.2 : DT Impulse Responces

• Impulse Responce (i.e. h[n]) : Output excited by input δ[n]
• If u[n] is Initially Relaxed
• (The filter is causal) ↔ (h[n]=0 for n<0)

### Section 3.2.1 : FIR and IIR Systems

• IIR (Infinite Impulse Responce) : h[n] is non-zero for infinitely many n
• FIR (Finite Impulse Responce) : h[n] is non-zero for finitely many n
• FIR Length : The range over n for which h[n] is non-zero
• Memoryless System : Always FIR of Length 1
• N unit time-delay : Always FIR of Length 1+N
• Length N : System must have memory of u[n] at N different 'n's

### Section 3.3 : DT LTI Systems - Discrete Convolutions

• System Initially Relaxed at:
• t0 = 0 : y[n] = K = 0Σ u[n-k]h[k]
• t0=–∞ : y[n] = K=–∞Σ u[n-k]h[k]

### Section 3.3.1 : Underlying Procedure of Discrete Convolutions

• h[n] * u[n]:
• Flip h[k] to h[-k], Shift h[-k] to h[n-k], Multiply h[n-k] and u[k] for all k, Sum to get y[n]
• Sum of Products of u[k] with the Shift of the Flip of h[k]
• u[k] DOT Shift(Flip(h[k]))

### Section 3.4 : DT LTI Lumped Systems - Difference Equations

• DT LTI Lumped Systems : can be written as Difference Equations
• DT LTI Lumped Systems : can be written as Convolution
• Convolution : can be written as Difference Equations

### Section 3.4.1 : Setting Up Difference Equations

• Difference Equation can be setup directly from model
• E.G. Bank Account y[n] = a * y[n-1] + u[n] : can be written as y[n] - a * y[n-1] = u[n]

### Section 4.1 : Frequency Spectra of CT Signals

•  Input\Output DT CT DT DFT or FFT DTFT CT Fourier Series Fourier Transform
• t : (–∞, +∞)
• x(t) : Time-domain
• X(ω) : Frequency-domain

### Section 4.1.1 : Orthogonality of Complex Exponentials

• φm(t) : ejmω₀t
• φk*(t) : e–jmω₀t
• <P> φm(t) dt : p·δ[m]
• <P> φm(t)φk*(t) dt : p·δ[m-k]

### Section 4.2 : Fourier Series of Periodic Signals - Frequency Components

• x(t) : m=–∞Σ cmejmω₀t (Synthesis Equation)
• cm : (1/P) ∫<P> x(t)e-jmω₀tdt (Analysis Equation)

### Section 4.2.1 : Properties of Fourier Series Components

• If x(t) is real
• c–m : c*m (Conjugate Symmetry)

### Section 5.1 : Introduction

• ℱ(DT) : ℱ(CT(DT))

• H : ℱh
• · : ℒ*