The unit impulse in continuous time is the dirac delta function
δ(t)=limΔ→0δΔ(t)δΔ(t)={Δ1,0if 0≤t<Δelse.
f(t)δ(t)=f(0)δ(t)
f(t)δ(t−τ)=f(τ)δ(t−τ)
δ(at)=∣a∣1δ(t)
Definition 6
The unit step is defined as
u[n]={10if n≥0else.
Signal transformations
Signals can be transformed by modifying the variable.
x(t−τ): Shift a signal right by τ steps.
x(−t): Rotate a signal about the t=0
x(kt): Stretch a signal by a factor of k
These operations can be combined to give more complex transformations. For example, y(t)=x(τ−t)=x(−(t−τ)) flips x and shifts it right by τ timesteps. This is equivalent to shifting x left by τ 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]
Definition 8
The convolution of two signals in continuous time
(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[n]∗δ[n−N]=x[n−N]
(x∗h)[n]=(h∗x)[n]
x∗(h1+h2)=x∗h1+x∗h2
x∗(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 ∃a bounded input that makes the output unbounded.
Definition 13
A system is time-invariant if the original input x(t) is transformed to y(t), then x(t−τ) is transformed to y(t−τ)
Definition 14
A system f(x) is linear if and only if for the signals y1(t)=f(x1(t)),y2(t)=f(x2(t)), then scaling (f(ax1(t))=ay(t) and superposition (f(x1(t)+x2(t))=y1(t)+y2(t)) hold.
Notice: The above conditions on linearity require that x(0)=0 because if a=0, then we need y(0)=0 for scaling to be satisfied
Definition 15
The impulse response of a system f[x] is 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]decays to zero in a finite amount of time
Definition 17
A system has an Infinite Impulse Response (IIR) if 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)
Definition 18
The frequency response of a system is how a system responds to a purely oscillatory signal