Introduction to Signals and Systems

Types of Signals
Definition 1
A signal is a function of one or more variables
Definition 2
A continuous signal x(t)x(t)x(t)
 maps R→R\mathbb{R} \rightarrow \mathbb{R}R→R
​
Definition 3
A discrete signal x[n]x[n]x[n]
 maps Z→R\mathbb{Z} \rightarrow \mathbb{R}Z→R
​
Properties of the Unit Impulse
Definition 4
The unit impulse in discrete time is defined as
​δ[n]={1if n=00else \delta[n] = \begin{cases} 1 & \text{if } n = 0\\ 0 & \text{else } \end{cases}δ[n]={10​if n=0else ​
​
​f[n]δ[n]=f[0]δ[n]f[n]\delta[n] = f[0]\delta[n]f[n]δ[n]=f[0]δ[n]
​
​f[t]δ[n−N]=f[N]δ[n−N]f[t]\delta[n-N] = f[N]\delta[n-N]f[t]δ[n−N]=f[N]δ[n−N]
​
Definition 5
The unit impulse in continuous time is the dirac delta function
​δ(t)=limΔ→0δΔ(t)δΔ(t)={1Δ,if 0≤t<Δ0else.\delta(t)=lim_{\Delta\rightarrow 0}\delta_{\Delta}(t) \qquad \delta_{\Delta}(t)=\begin{cases} \frac{1}{\Delta}, & \text{if } 0 \leq t < \Delta \\ 0 & \text{else.} \end{cases}δ(t)=limΔ→0​δΔ​(t)δΔ​(t)={Δ1​,0​if 0≤t<Δelse.​
​
​f(t)δ(t)=f(0)δ(t)f(t)\delta(t) = f(0)\delta(t)f(t)δ(t)=f(0)δ(t)
​
​f(t)δ(t−τ)=f(τ)δ(t−τ)f(t)\delta(t-\tau) = f(\tau)\delta(t-\tau)f(t)δ(t−τ)=f(τ)δ(t−τ)
​
​δ(at)=1∣a∣δ(t)\delta(at) = \frac{1}{|a|}\delta(t)δ(at)=∣a∣1​δ(t)
​
Definition 6
The unit step is defined as
​u[n]={1if n≥00else.u[n] = \begin{cases} 1 & \text{if } n \geq 0\\ 0 & \text{else.} \end{cases}u[n]={10​if n≥0else.​
​
Signal transformations
Signals can be transformed by modifying the variable.
​x(t−τ)x(t - \tau)x(t−τ)
: Shift a signal right by τ\tauτ
 steps.
​x(−t)x(-t)x(−t)
: Rotate a signal about the t=0t=0t=0
​
​x(kt)x(kt)x(kt)
: Stretch a signal by a factor of kkk
​
These operations can be combined to give more complex transformations. For example, y(t)=x(τ−t)=x(−(t−τ))y(t) = x(\tau - t) = x(-(t-\tau))y(t)=x(τ−t)=x(−(t−τ))
 flips xxx
 and shifts it right by τ\tauτ
 timesteps. This is equivalent to shifting xxx
 left by τ\tauτ
 timesteps and then flipping it.
Convolution
Definition 7
The convolution of two signals in discrete time
​(x∗h)[n]=∑k=−∞∞x[k]h[n−k](x*h)[n] = \sum_{k=-\infty}^{\infty}{x[k]h[n-k]}(x∗h)[n]=∑k=−∞∞​x[k]h[n−k]
​
Definition 8
The convolution of two signals in continuous time
​(x∗h)(t)=∫−∞∞x(τ)h(t−τ)dτ(x*h)(t) = \int_{-\infty}^{\infty}{x(\tau)h(t-\tau)d\tau}(x∗h)(t)=∫−∞∞​x(τ)h(t−τ)dτ
​
While written in discrete time, these properties apply in continuous time as well.
​(x∗δ)[n]=x[n](x*\delta)[n] = x[n](x∗δ)[n]=x[n]
​
​x[n]∗δ[n−N]=x[n−N]x[n]*\delta[n-N]=x[n-N]x[n]∗δ[n−N]=x[n−N]
​
​(x∗h)[n]=(h∗x)[n](x*h)[n] = (h*x)[n](x∗h)[n]=(h∗x)[n]
​
​x∗(h1+h2)=x∗h1+x∗h2x * (h_1 + h_2) = x*h_1 + x*h_2x∗(h1​+h2​)=x∗h1​+x∗h2​
​
​x∗(h1∗h2)=(x∗h1)∗h2x * (h_1 * h_2) = (x * h_1) * h_2x∗(h1​∗h2​)=(x∗h1​)∗h2​
​
Systems and their properties
Definition 9
A system is a process by which input signals are transformed to output signals
Definition 10
A memoryless system has output which is only determined by the input's present value
Definition 11
A causal system has output which only depends on input at present or past times
Definition 12
A stable system produces bounded output when given a bounded input. By extension, this means an unstable system is when ∃\exists∃
a bounded input that makes the output unbounded.
Definition 13
A system is time-invariant if the original input x(t)x(t)x(t)
 is transformed to y(t)y(t)y(t)
, then x(t−τ)x(t-\tau)x(t−τ)
 is transformed to y(t−τ)y(t-\tau)y(t−τ)
​
Definition 14
A system f(x)f(x)f(x)
 is linear if and only if for the signals y1(t)=f(x1(t)),y2(t)=f(x2(t))y_1(t) = f(x_1(t)), y_2(t) = f(x_2(t))y1​(t)=f(x1​(t)),y2​(t)=f(x2​(t))
, then scaling (f(ax1(t))=ay(t)f(a x_1(t)) = a y(t)f(ax1​(t))=ay(t)
 and superposition (f(x1(t)+x2(t))=y1(t)+y2(t)f(x_1(t) + x_2(t)) = y_1(t) + y_2(t)f(x1​(t)+x2​(t))=y1​(t)+y2​(t)
) hold.
Notice: The above conditions on linearity require that x(0)=0x(0) = 0x(0)=0
 because if a=0a = 0a=0
, then we need y(0)=0y(0) = 0y(0)=0
 for scaling to be satisfied
Definition 15
The impulse response of a system f[x]f[x]f[x]
 is h[n]=f[δ[n]]h[n] = f[\delta[n]]h[n]=f[δ[n]]
, which is how it response to an impulse input.
Definition 16
A system has a Finite Impulse Response (FIR) if h[n]h[n]h[n]
decays to zero in a finite amount of time
Definition 17
A system has an Infinite Impulse Response (IIR) if h[n]h[n]h[n]
does not decay to zero in a finite amount of time
Exponential Signals
Exponential signals are important because they can succinctly represent complicated signals using complex numbers. This makes analyzing them much easier.
​x(t)=est,x[n]=zn(s,z∈C)x(t) = e^{st}, x[n] = z^n (s, z \in \mathbb{C})x(t)=est,x[n]=zn(s,z∈C)
​
Definition 18
The frequency response of a system is how a system responds to a purely oscillatory signal
EE120
The Fourier Series
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