Signals can be transformed by modifying the variable.
​
x(t−τ)
: Shift a signal right by
Ï„
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
Ï„
timesteps. This is equivalent to shifting
x
left by
Ï„
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