Unit Impulse Function
Lesson #2
2CT.2,4,
3CT.2
Appendix A
BME 333 Biomedical Signals and Systems
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Complex Numbers
•  Constants:
s = a + jb
Rectangular Form
Complex
Plane
a is called the Real part of s
b is called the Imaginary part of s
= a 2 + b2 e
Imaginary
axis
b
tan-1(b/a)
a
−1
j tan ( b )
a
= a 2 + b 2 ∠ tan −1 ( b ) Polar Form
a
•  Functions:
Example : e
jω t
Imaginary
axis
Real axis
Rotating
Unit Vector
at rate w
= cos ω t + j sin ω t
jθ
(recall: e = cosθ + j sin θ )
BME 333 Biomedical Signals and Systems
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Real axis
18
Complex Exponential Function as a function of
time
j 2π (1) t
j 2πt
z
(
t
)
=
1
e
=
e
= cos 2πt + j sin 2πt
•  Let’s look at this
t=2/8 seconds
t=8/8 seconds
arg(z(t))=2π x8/8 = 2π ; z(t)= 1+ j0
arg(z(t))=2π x2/8= π /2; z(t)= 0 + j1
t=3/8 seconds
t=1/8 seconds
Im{z}
arg(z(t))=2π x3/8 = 3 π /4;
arg(z(t))=2π x1/8=π/4; z(t)=0.707+j 0.707
z(t)= -0.707+ j0.707
t=0 seconds
t=4/8 seconds
45o
arg(z(t))=2π x0=0; z(t)=1+ j0
arg(z(t))=2π x4/8 = π; z(t)= -1+ j0
Re{z}
t=5/8 seconds
t=7/8 seconds
arg(z(t))=2π x5/8 = 5π /4;
arg(z(t))=2π x7/8= 7π /4;
z(t)= -0.707 - j0.707
t=6/8 seconds
z(t) = 0 .707- j0.707
arg(z(t))=2π x6/8 = 3π /2; z(t) = 0 - j
BME 333 Biomedical Signals and Systems
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Phasor Representation of a Complex Exponential
Signal
•  Using the multiplication rule, we can rewrite
the complex exponential signal as
z (t ) = Ae j (ωot +φ ) = Ae jωot e jφ = Ae jφ e jωot = Xe jωot = Xe j 2π Fot
where X is a complex number equal to
X = Ae jφ
•  X is complex amplitude of the complex
exponential signal and is also called a phasor
BME 333 Biomedical Signals and Systems
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Phasors
• 
• 
Note that the real sinusoidal function
f(t)=Acos (ωt+ϕ)
can be represented by a complex function
f(t)=A cos (ωt+ϕ) = Re[Ae j(ωt+ϕ)]
Let’s represent this function by a phasor which is
its magnitude and phase angle:
f (t ) = A cos(ωt + φ ) = Re[ Ae j (ωt +φ ) ] = Re[ Ae jφ e jωt ] ⇒ A∠φ
• 
Therefore, we can use phasors to represent
complex functions which makes it easy to solve
and calculate system solutions
BME 333 Biomedical Signals and Systems
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Example Using ODE with Trigonometry
•  Let’s calculate the current I(t) assuming V(t)= A cos ωt
dI (t )
= V (t ) = A cos ωt
dt
Use Trigonometric functions
RI (t ) + L
R
dI (t )
= − I ω sin(ωt + θ )
dt
RI cos(ωt + θ ) − I ω L sin(ωt + θ ) = A cos ωt
Let I (t ) = I cos(ωt + θ );
I(t)
V(t)
L
To solve for I and θ , use the identities:
cos( A + B) = cos A cos B − sin A sin B; sin( A + B) = sin A cos B + cos A sin B
RI [cos ωt cos θ − sin ωt sin θ ] − I ω L[sin ωt cos θ + cos ωt sin θ ] = A cos ωt
0
sin θ
−ω L
−ω L
= tan θ =
⇒ θ = tan −1 (
)
cos θ
R
R
A
A
RI cos θ − I ω L sin θ = A ⇒ I =
=
R
−ω L
R cos θ − ω L sin θ R
− ω L(
)
2
2
2
2
R + (ω L)
R + (ω L)
RI [− sin θ ] − I ω L cos θ = 0 ⇒ R[sin θ ] = −ω L cos θ ⇒
I=
A
=
R + (ω L) 2
2
R + (ω L)
2
I (t ) =
A
R + (ω L)
2
R
θ
-ωL
R2 + (ω L)2
2
2
A
R 2 + (ω L) 2
cos(ωt + tan −1 (
−ω L
))
R
MESSY !!!!
BME 333 Biomedical Signals and Systems
- J.Schesser
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Example Using ODE with Complex Exponentials
•  Let’s calculate the current I(t) assuming
R
V(t)= A cos ωt
dI (t)
= V (t) = Acos ω t
dt
Use complex exponent functions
RI (t) + L
V(t)
I(t)
Let I (t) = I cos(ω t + θ ) = ℜe{Ie jθ e jω t };Let V (t) = Acos(ω t) = ℜe{Ae jω t };
L
0
dI (t)
= jω Ie jθ e jω t
dt
RIe jθ e jω t + jω LIe jθ e jω t = Ae jω t
RIe jθ + jω LIe jθ = A
A
Ie =
=
R + jω L
A
jθ
I (t) = ℜe{
R + (ω L)
2
A
R 2 + (ω L) 2
e
2
− j tan −1
e
ωL
R
− j tan −1
ωL
R
e jω t } =
A
R 2 + (ω L) 2
cos(ω t − tan −1
BME 333 Biomedical Signals and Systems
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ωL
)
R
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A Special Function – Unit Impulse Function
•  The unit impulse function, δ(t), also known as the
Dirac delta function, is defined as:
δ(t)
δ(t) = 0 for t ≠ 0;
= undefined for t = 0
and has the following special property:
0
-100
∞
∫ f (t)δ (t −τ )dt = f (τ )
−∞
∞
∴ ∫ δ (t)dt =1
−∞
BME 333 Biomedical Signals and Systems
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-50
-25
-1
0
1
25
50
100
24
Unit Impulse Function Continued
•  A consequence of the delta function is that
it can be approximated by a narrow pulse
as the width of the pulse approaches zero
while the area under the curve = 1
δ(t)
lim δ (t) ≈1/ε for -ε / 2 < t < ε / 2; = 0 otherwise.
ε →0
10
1
0.5
-1
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-.5 -.05 .05
.5
1
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Unit Impulse Function Continued
∞
∫
f (t)δ (t − τ ) dt
−∞
Let's approximate δ (t − τ ) with a pulse of height
∞
∫
f (t)δ (t − τ ) dt ≈
∫
τ −ε 2
−∞
τ
τ +ε 2
1
and width ε
ε
1/ ε
1
f (t) dt
ε
If we take the limit of this integral as ε → 0,
τ - ε /2
τ
τ + ε/2
the approximation integral approaches the original integral
∞
∫
−∞
τ +ε 2
f (t)δ (t − τ ) dt = lim
∫
ε →0 τ −ε 2
1
1
f (t) dt → lim f (τ ) ε = f (τ ),
ε
ε
ε →0
since as ε → 0, the integral is zero except at t = τ
BME 333 Biomedical Signals and Systems
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Uses of Delta Function
•  Modeling of electrical, mechanical, physical
phenomenon:
–  point charge,
–  impulsive force,
–  point mass
–  point light
BME 333 Biomedical Signals and Systems
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Another Special Function – Unit Step
Function
•  The unit step function, u(t) is defined as:
u(t) = 1 for t ≥ 0;
1
= 0 for t < 0.
t
and is related to the delta function as
follows:
t
u(t ) = ∫−∞ δ (τ )dτ
BME 333 Biomedical Signals and Systems
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Integration of the Delta Function
•  δ(t)
•  u(t)
•  tu(t)
.
.
.
• 
u(t)
tu(t)
t 2 u(t)
2!
t n u(t)
n!
1st order
2nd order
nth order
BME 333 Biomedical Signals and Systems
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Signal Representations using the Unit Step
Function
•  x(t) =
e-σt
1.2
cos(ωt)u(t)
1
0.8
0.6
0.4
0.2
-0.5
0
-0.2 0
0.5
1
1.5
2
2.5
3
-0.4
-0.6
-0.8
•  x(t) = t u(t) – 2 (t-1)u(t-1) + (t-2) u(t-2)
6
tu(t)
4
(t-2)u(t-2)
2
0
-2
-4
0
1
2
3
4
5
6
x(t)
-2(t-1)u(t-1)
-6
BME 333 Biomedical Signals and Systems
- J.Schesser
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Homework
• 
Complex numbers
– 
– 
– 
– 
– 
• 
Convert 1+j1 to its magnitude/angle representation (phasor)
Convert 1/(1+j1) to a phasor
Draw ejωt and ej(ωt+α) in the complex plane
For the series R-L circuit in class, calculate the voltage across the
inductor.
Appendix A.4, A.7
Unit Impulse and Unit Step Functions
– 
– 
– 
– 
– 
Using unit step functions, construct a single pulse of magnitude
10 starting at t=5 and ending at t=10.
Repeat problem 1) with 2 pulses where the second is of
magnitude 5 starting at t=15 and ending at t=25.
Is the unit step function a bounded function?
Is the unit impulse function a bounded function?
2CT.2.4a,b
BME 333 Biomedical Signals and Systems
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