Lecture II: Continuous-Time and Discrete-Time
Signals
Maxim Raginsky
BME 171: Signals and Systems
Duke University
August 29, 2008
Maxim Raginsky
Lecture II: Continuous-Time and Discrete-Time Signals
This lecture
Plan for the lecture:
1
Review: complex numbers
2
Continuous-time signals
unit step and unit ramp
unit impulse
transformations of time
3
Discrete-time signals
unit step
unit impulse
4
Periodic continuous-time and discrete-time signals
Maxim Raginsky
Lecture II: Continuous-Time and Discrete-Time Signals
Review: complex numbers
Rectangular form: s = a + jb,
a = Re(s), b = Im(s)
√
j = −1
Im s
b
θ
0
Polar form: s = rejθ
Euler’s formula: ejθ = cos θ + j sin θ
Complex conjugate: s∗ = a − jb = re−jθ
ss∗ = |s|2 = a2 + b2
Maxim Raginsky
a Re s
p
a2 + b 2
b
θ = tan−1
a
r = |s| =
Lecture II: Continuous-Time and Discrete-Time Signals
Unit step and unit ramp
Unit step:
u(t)
u(t) =
1
1, t ≥ 0
0, t < 0
t
0
Unit ramp:
r(t) =
0
Running integral representation:
Z
r(t) =
slo
1
t, t ≥ 0
0, t < 0
pe
=1
r(t)
1
t
t
u(τ )dτ
−∞
Maxim Raginsky
Lecture II: Continuous-Time and Discrete-Time Signals
Unit impulse
Unit impulse (aka Dirac delta-function):
δ(t)
1
2
(1)
δ(t) = 0 for t 6= 0
Ra
δ(t)dt = 1 for any a > 0
−a
t
0
The value of δ(t) at t = 0 is undefined; in particular, it is not +∞!
It is useful to think of δ(t) as an infinitesimally narrow pulse of unit area
centered around 0:
δε(t)
1/ε
δ(t) = lim δε (t),
area=1
ε→0
where
δε (t) =
1/ε, −ε/2 ≤ t ≤ ε/2
0,
|t| > ε/2
Maxim Raginsky
−ε/2
0
+ε/2
t
Lecture II: Continuous-Time and Discrete-Time Signals
The main property of the unit impulse
If x(t) is a signal that is continuous at t = 0, then
x(t)δ(t) = x(0)δ(t).
In particular,
Z
a
x(t)δ(t)dt = x(0)
for any 0 < a ≤ +∞.
−a
You can convince yourselves of this by approximating δ(t) with a pulse,
such as δε (t), and using the fact that, if ε is small enough, then
x(t) ≈ x(0)
for − ε/2 ≤ t ≤ ε/2.
1/ε
−ε/2
Maxim Raginsky
0
+ε/2
x(t)
t
Lecture II: Continuous-Time and Discrete-Time Signals
Transformations of time
Time reversal: x(t) −→ x(−t)
x(t)
Time shifts: x(t) −→ x(t − t1 )
x(-t)
0
t
0
x(t)
t
x(t-t1)
0
t
0
t1
t
Time scaling: x(t) −→ x(ct)
x(t)
0
x(t/2)
t0
t
Maxim Raginsky
0
x(2t)
2t0
t
0
t0/2
Lecture II: Continuous-Time and Discrete-Time Signals
t
Examples
Πτ (t) = u t +
τ
2
−u t−
τ
2
τ
2
Πτ(t)
1
-τ/2
Λτ (t) =
2
τ
t+
τ
2
τ/2
0
Πτ /2 t +
τ
4
−
2
τ
t
t−
Πτ /2 t −
τ
4
Λτ(t)
1
-τ/2
Maxim Raginsky
0
τ/2
t
Lecture II: Continuous-Time and Discrete-Time Signals
More examples
x(t) = 2(t + 2)Π1 (t + 1.5) −
2(t − 2)Π1 (t − 1.5) + 2Π2 (t)
x(t) = 2Π2.5 (t − 0.25)
x(t)
x(t)
2
2
-1
0
1.5
t
-2
x(t) =
P∞
k=−∞
-1
0
1
2
t
g(t − kπ), where g(t) = tΠπ (t)
x(t)
1
-π
0
π
−1
Maxim Raginsky
Lecture II: Continuous-Time and Discrete-Time Signals
Shifted unit impulse and the sifting property
Unit impulse located at t = t1 :
δ(t-t1)
(1)
t1
0
t
Example: neural spike trains
x(t)
x(t) =
K
P
k=1
0
δ(t − tk )
tk , 1 ≤ k ≤ K: spike times
interspike intervals tk+1 − tk : milliseconds
t
The sifting property of the unit impulse: for any signal x(t) that’s
continuous at t = t1 ,
Z ∞
x(t)δ(t − t1 )dt = x(t1 )
−∞
Maxim Raginsky
Lecture II: Continuous-Time and Discrete-Time Signals
Basic discrete-time signals
δ[n]
Discrete-time unit impulse:
1, n = 0
δ[n] =
0, n = ±1, ±2, . . .
-1 -2
0 1 2
n
0 1 2
n
u[n]
Discrete-time unit step:
1, n = 0, 1, 2, . . .
u[n] =
0, n = −1, −2, . . .
-1 -2
It is easy to see that
x[n]δ[n] =
x[0], n = 0
0,
n = ±1, ±2, . . .
Maxim Raginsky
u[n] =
∞
X
k=0
δ[n − k]
Lecture II: Continuous-Time and Discrete-Time Signals
Periodic continuous-time signals
x(t) is periodic if there exists a number T > 0, such that
x(t + T ) = x(t),
for all t.
Fundamental period: smallest positive T , such that the above holds.
Examples:
x(t)
period = 2π/ω
x(t)
period=π
1
t
0
-π
0
π
−1
sinusoid x(t) = A cos(ωt + θ)
Maxim Raginsky
triangular wave
Lecture II: Continuous-Time and Discrete-Time Signals
Sums of periodic signals
Suppose x1 (t) is periodic with period T1 and x2 (t) is periodic with
period T2 . Then
x(t) = x1 (t) + x2 (t)
is periodic if and only if there exist positive integers q and r, such that
rT1 = qT2 . Moreover, if r and q are relatively prime (i.e., have no
common multiple except 1), then T = rT1 is the fundamental period of
x(t).
Example:
x(t) = 5 cos(3πt + 1.2) − 8 sin(5πt − 4) is periodic
x(t) = 5 cos(3πt + 1.2) − 8 sin(5t − 4) is not periodic
Maxim Raginsky
Lecture II: Continuous-Time and Discrete-Time Signals
Periodic discrete-time signals
x[n] is periodic if there exists a positive integer T , such that
x[n + T ] = x[n],
for all n.
Fundamental period: smallest positive integer T , such that the above
holds.
x[n]
0
n
fundamental period = 6
Example: x[n] = A cos(Ωn + θ) is periodic if and only if there are
positive integers q and r, such that Ω = 2πq/r (in other words, if Ω is a
rational multiple of 2π).
Maxim Raginsky
Lecture II: Continuous-Time and Discrete-Time Signals