Olin College of Engineering
ENGR2410 – Signals and Systems
Lecture 5 Reference
From Fourier series to Fourier transform
The integral of f (x) is the limit of the sum of the area of rectangles under the function as
they become narrower (∆x → 0) and denser (n → ∞),
∫
∑
xb /∆x
xb
f (x)dx = lim
∆x→0
xa
f (n∆x)∆x
n=xa /∆x
As derived in the last lecture, an even square wave v(t) with period T and pulse width 2T1
can be expressed as sum of complex exponentials such that
(
)
∞
∑
2π
T1
j 2π
nt
v(t) =
cn e T ,
cn = 2V sinc
nT1
T
T
n=−∞
with each n corresponding to a frequency ω =
then ∆ω → 0. The original v(t) becomes
v(t) =
2π
n.
T
Define ∆ω ,
2π
T
such that if T → ∞,
∞
∑
∆ω
2V T1 sinc (∆ωnT1 ) ej∆ωnt
|
{z
}
2π
|{z}
n=−∞
cn
1/T
π
. As ∆ω → 0,
The first zero of the sinc function occurs when ∆ωnT1 = π such that n = ∆ωT
1
n → ∞, but the first zero does not change from ∆ωn = π/T1 . In general, ω = ∆ωn. As
∆ω → 0, the summation converges to
∫ ∞
dω
v(t) =
2V T1 sinc (ωT1 ) ejωt
{z
}
2π
ω=−∞ |
V (jω)
LTI system response in the frequency domain
In general, the Fourier transform of x(t) is X(jω) = F {x(t)}, where
∫ ∞
F {x(t)} =
x(t)e−jωt dt
t=−∞
Similarly, the inverse Fourier transform of X(jω) is x(t) = F −1 {X(jω)}, where
∫ ∞
dω
−1
F {X(jω)} =
X(jω)ejωt
2π
ω=−∞
Equivalently, we can write x(t)
F
⇐⇒
X(jω).
The response to an exponential ejωt of any LTI system with transfer function H(jω) is given
by
ejωt → LT I → H(jω)ejωt
If we can express the input x(t) to a system as an integral of complex exponentials, the
output y(t) is equal to the same integral of complex exponentials scaled by the
transfer function. Make sure you understand this last sentence clearly.
∫ ∞
∫ ∞
jωt
x(t) =
X(jω)e dt → H(jω) → y(t) =
H(jω)X(jω) ejωt dt
{z
}
−∞
−∞ |
Y (jω)
In system notation,
x(t)


Fy
−−−→
LTI
−−−→
y(t)
x
 −1
F
X(jω) −−−→ H(jω) −−−→ Y (jω)= X(jω)H(jω)
2
Impulse and impulse response
The impulse response, as we will see shortly, is the inverse Fourier transform of a given
system (h(t) = F −1 {H(jω)}). We will now show that the impulse response depends only
of the area of the impulse, regardless of its shape. In particular, the differential equation
describing the behavior of the circuit below is given by
v̇ +
1
1
v = i(t)
RC
C
{
The input current i(t) is given by the piecewise function i(t) =
I0
0
0 < t < t0
shown
otherwise
below.
{
The output voltage is v(t) =
I0 R(1 − e−t/τ )
I0 R(1 − e−t0 /τ )e−(t−t0 )/τ
0 < t < t0
,
t > t0
τ = RC
The integral of i(t) is equal to the charge Q0 = I0 t0 . If we scale the current I0 such that the
charge Q0 remains constant, the solution becomes

Q

 0 R(1 − e−t/τ )
0 < t < t0
t0
v(t) =
Q

 0 R(1 − e−t0 /τ )e−(t−t0 )/τ
t > t0
t0
If we let t0 → 0, i(t) = Q0 δ(t), where δ(t) is the unit impulse. In the same limit, the output
voltage v(t) is the impulse response, equal to

Q

 0 R(1 − e−t/τ )
0 < t < t0
t0
v(t) = lim
t0 →0  Q0

R(et0 /τ − 1)e−t/τ
t > t0
t0
(
)
Q0
t0
Q0 t0 −t/τ
Q0 −t/τ
v(t) ≈
R 1 + − 1 e−t/τ =
e
, t>0
R
e
=
t0
τ
C
t0 RC
3
Properties of the impulse
Most properties of the impulse can be shown by constructing appropriate functions and then
taking the limit. For example, in order to show that the unit impulse is the derivative of the
unit step, δ(t) = du(t)
, use the function u0 (t) shown below.
dt
Its derivative is the function δ0 (t),
In the limit when t0 → 0, u0 (t) → u(t) and δ0 (t) → δ(t). The impulse δ(t) may be defined
as
∫ 0+
−
+
δ = 0, (−∞, 0 ) ∪ (0 , ∞);
δ(t) = 1
0−
The impulse can be scaled and shifted, leading to the “picking” property,
∫ ∞
f (t)δ(t − t0 )dt = f (t0 )
−∞
This property allows us to compute the Fourier transform of the impulse,
∫ ∞
F {δ(t)} =
δ(t)e−jωt dt = ejω(0) = 1
−∞
Using the inverse Fourier transform, the previous transform implies that the impulse is the
integral of all complex exponentials, equally weighted,
∫ ∞
dω
δ(t) =
1 · ejωt
2π
−∞
4
Finally, since F {δ(t)} = 1, the frequency content of the impulse response is 1 · H(jω).
Therefore, h(t) = F −1 {H(jω)} is the impulse response of a system with transfer function
H(jω). In system notation,
δ(t) −−−→ H(jω) −−−→


Fy
1
h(t)
x
 −1
F
−−−→ H(jω) −−−→ H(jω)
The behavior of a system can be fully characterized equally by using a differential equation,
a transfer function, a Bode plot, or the impulse response.
5