MSE280s Chapter1_Signals_and_Systems.pdf

MSE 280 Linear Systems
MSE280: Linear Systems
Chapter 1
Signals and Systems
Mohammad Narimani, Ph.D. P.Eng.
Lecturer
School of Mechatronic Systems Engineering
Simon Fraser University

MSE 280 Linear Systems 2
Objectives
▪ Review of continuous and discontinuous (discrete)
signals
▪ Introduction of Signal Energy and Power
▪ An Introduction to “Transformations of the
independent variable”
▪ An Introduction to “Exponential and sinusoidal
signals”

Review >> Signals
▪ CT and DT signals
▪ CT signals are denoted by symbol t and the independent
variable are enclosed in bracket (.)
▪ DT signals are denoted by symbol n and the independent
variable are enclosed in bracket [.]

Review >> Signals
▪ A DT signal x[n] may represent a phenomenon for which the
independent variable is inherently discrete.
Example:
▪ A DT signal x[n] may represent successive samples of a
phenomenon for which the independent variable is continuous.
Example: the processing of speech on a digital computer
requires the use of a discrete time sequence representing the
values of the continuous-time speech signal at discrete points
of time.

Complex Numbers
▪ We are interested in the general complex signals:
x(t) ∈ ℂ and x[n] ∈ ℂ
where the set of complex numbers is defined as
ℂ = 𝑧 𝑧 = 𝑥 + 𝑗𝑦, 𝑥, 𝑦 ∈ ℝ, 𝑗 = −1
▪ A complex number 𝑧 can be represented in Cartesian form as
𝑧 = 𝑥 + 𝑗𝑦
or in polar form as
𝑧 = 𝑟𝑒𝑗𝜃
▪ Euler's Formula
The relation between 𝑥, 𝑦, 𝑟, and 𝜃 is given by
ቊ
𝑥 = 𝑟 cos 𝜃
𝑦 = 𝑟 sin 𝜃
and ቐ
𝑟 = 𝑥2 + 𝑦2
𝜃 = tan−1 𝑦
𝑥
𝑒𝑗𝜃 = cos 𝜃 + 𝑗 sin 𝜃

Signal Energy and Power
▪ If v(t) and i(t) are respectively the voltage and current across a resistor with
resistance R , then the instantaneous power is
p(t)=v(t)i(t)=
1
𝑅
𝑣 𝑡 2
The total energy expended over the time interval 𝑡1 ≤ 𝑡 ≤ 𝑡2 is
න
𝑡1
𝑡2
𝑝 𝑡 𝑑𝑡 = න
𝑡1
𝑡2 1
𝑅
𝑣 𝑡 2𝑑𝑡
and the average power over this time interval is
1
𝑡2 − 𝑡1
න
𝑡1
𝑡2
𝑝 𝑡 𝑑𝑡 =
1
𝑡2 − 𝑡1
න
𝑡1
𝑡2 1
𝑅
𝑣 𝑡 2
𝑑𝑡

▪ For any continuous-time signal x(t) or any discrete-time signal x[n] , the total
energy over the time interval 𝑡1 ≤ 𝑡 ≤ 𝑡2 in a continuous-time signal x(t) is defined
න
𝑡1
𝑡2
𝑥 𝑡 2𝑑𝑡
where 𝑥 denotes the magnitude of the (possibly complex) number x .
The time-averaged power is
1
𝑡2 − 𝑡1
න
𝑡1
𝑡2
𝑥 𝑡 2
𝑑𝑡
▪ Similarly the total energy in a discrete-time signal x[n] over the time interval 𝑛1 ≤
𝑛 ≤ 𝑛2 is defined as
෍
𝑛1
𝑛2
x[n] 2
And the average power is
1
𝑛2 − 𝑛1 + 1
෍
𝑛1
𝑛2
x[n] 2

▪ The energy in signals over an infinite time interval, that is
𝐸∞ = lim
𝑇→∞
න
−𝑇
𝑇
𝑥 𝑡 2
𝑑𝑡 = න
−∞
+∞
𝑥 𝑡 2
𝑑𝑡
and in discrete time
𝐸∞ = lim
𝑁→∞
෍
−𝑁
𝑁
x[n] 2
= ෍
−∞
+∞
x[n] 2
▪ The time-averaged power over an infinite interval
𝑃∞ = lim
𝑇→∞
1
2𝑇
න
−𝑇
𝑇
𝑥 𝑡 2
𝑑𝑡
and in discrete time
𝑃∞ = lim
𝑁→∞
1
2𝑁 + 1
෍
−𝑁
𝑁
x[n] 2
▪ Energy Signal: signals with finite total energy, 𝐸∞ < ∞ and zero average power
▪ Power Signal: with finite average power 𝑃∞. If 𝑃∞ > 0 , then 𝐸∞ = ∞. An example is the
signal x[n] = 4, it has infinite energy, but has an average power of 𝑃∞ = 16.

Example:

1.2 Transformations of the independent variable
Time Shift: For any 𝑡0 ∈ ℝ and 𝑛0 ∈ ℤ, time shift is an operation defined as
𝑥 𝑡 → 𝑥(𝑡 − 𝑡0)
𝑥[𝑛] → 𝑥[𝑛 − 𝑛0]
▪ If 𝑡0 > 0 (n0 > 0), the time shift is known as “delay”. If 𝑡0 < 0 (𝑛0 < 0), the time
shift is known as “advance”.

▪ Example: Obtain time-shifted version of 𝑥(𝑡 − 2) shown in the picture.

Time Reversal: Time reversal is dened as
𝑥 𝑡 → 𝑥(−𝑡)
𝑥[𝑛] → 𝑥[−𝑛]

▪ Example: Obtain 𝑥(−𝑡), where 𝑥(𝑡) is shown in the picture about 𝑡 = 0.

Time Scaling: Time scaling is the operation where the time variable t is multiplied by a
constant 𝑎:
𝑥 𝑡 → 𝑥(𝑎𝑡)
If 𝑎 > 1, the time scale of the resultant signal is “decimated” (speed up). If 0 < 𝑎 < 1,
the time scale of the resultant signal is “expanded” (slowed down).

▪ Example: Obtain 𝑥(2𝑡) for the 𝑥(𝑡) shown in the picture.

Decimation and Expansion: Decimation and expansion are standard discrete-time signal
processing operations.
Decimation is defined as
𝑦𝐷 𝑛 = 𝑥[𝑀𝑛]
for some integers M. M is called the decimation factor.
Expansion is defined as
𝑦𝐸 𝑛 = ቐ
𝑥
𝑛
𝐿
, 𝑛 = integer multiple of 𝐿
0, otherwise
𝐿 is called the expansion factor.

Example:
𝑀 = 2 and 𝐿 = 2

Combination of Operations: In general, linear operation (in time) on a signal 𝑥(𝑡) can
be expressed as 𝑦 𝑡 = 𝑥 𝑎𝑡 − 𝑏 , 𝑎, 𝑏 ∈ ℝ.
There are two methods to describe the output signal 𝑦 𝑡 = 𝑥 𝑎𝑡 − 𝑏 :
▪ Method A: “Shift”, then “Scale” (Recommended)
▪ Method B: “Scale”, then “Shift”
▪ Example: For the signal 𝑥(𝑡) shown in Figure below, sketch 𝑥(3𝑡 − 5).

▪ Example: For the signal 𝑥(𝑡) shown in Figure below, sketch 𝑥(2 − 𝑡).
Note: 𝑥(−(𝑡 − 𝑎)) is the flipped version of 𝑥 𝑡 − 2 around the value of ‘a’ (not
around zero).

1.2.2 Periodic Signals
▪ A periodic continuous-time signal 𝑥(𝑡) has the property that there is a positive value
of 𝑇 for which
𝑥 𝑡 = 𝑥(𝑡 + 𝑇) for all t
The fundamental period 𝑇0 of 𝑥(𝑡) is the smallest positive value of T for which the
above equation holds.
▪ A discrete-time signal 𝑥[𝑛] is periodic with period 𝑁 , where 𝑁 is an integer, if it is
unchanged by a time shift of 𝑁.
𝑥[𝑛] = 𝑥[𝑛 + 𝑁] for all n
The fundamental period 𝑁0 is the smallest positive value of 𝑁 for which the above
equation holds.
g

▪ Example: Consider the signal 𝑥(𝑡) = sin(𝜔0𝑡), 𝜔0 > 0. It can be shown
that 𝑥(𝑡) = 𝑥(𝑡 + 𝑇), where 𝑇 = 𝑘
2𝜋
𝜔0
for any 𝑘 ∈ ℤ+:
𝑥(𝑡 + 𝑇) = sin(𝜔0 𝑡 + 𝑇) = sin(𝜔0 𝑡 + 𝑘
2𝜋
𝜔0
)
= sin 𝜔0𝑡 + 2𝜋𝑘
= sin 𝜔0𝑡 + 2𝜋𝑘 = 𝑥(𝑡)

▪ Example: Determine the fundamental period of the following signals:
(a) 𝑒𝑗
3𝜋𝑡
5
(b) 𝑒𝑗
3𝜋𝑛
5

▪ Example: Determine the fundamental period of the following signals:
(a) 𝑥 𝑡 = cos(
𝜋𝑡2
8
)
(b) 𝑥[𝑛] = cos(
𝜋𝑛2
8
)

1.2.3 Even and Odd Signals
▪ Even and odd signal: A continuous-time signal 𝑥(𝑡) is even if
𝑥 −𝑡 = 𝑥 𝑡
and it is odd if
𝑥(−𝑡) = −𝑥(𝑡)
A discrete-time signal 𝑥[𝑛] is even if
𝑥[−𝑛] = 𝑥[𝑛]
and odd if
𝑥[−𝑛] = −𝑥[𝑛]
▪ Remark: The all-zero signal is both even and odd. Any other signal cannot be
both even and odd, but may be neither. The following simple example illustrate
these properties.
▪ Examples:
𝑥(𝑡) = 𝑡2
− 40
𝑥(𝑡) = 0.1𝑡3
𝑥 𝑡 = 𝑒0.4𝑡

1.2.3 Even and Odd Signals
▪ Decomposition Theorem: Every continuous-time signal x(t) can be expressed as:
𝑥 𝑡 = 𝑦 𝑡 + 𝑧(𝑡)
where 𝑦(𝑡) is even, and 𝑧(𝑡) is odd.
Proof:
𝐸𝑉{𝑥 𝑡 } =
1
2
[𝑥 𝑡 + 𝑥 −𝑡 ]
and
𝑂𝐷{𝑥 𝑡 } =
1
2
[𝑥 𝑡 − 𝑥 −𝑡 ]
▪ Example:
= +
𝑦[𝑛]
𝑦[𝑛]
𝑧[𝑛]
z[𝑛]

1.3 Exponential and sinusoidal signals
1.3.1 Continuous-time complex exponential and sinusoidal signals
Continuous-time complex exponential signal
𝑥(𝑡) = 𝐶𝑒𝑎𝑡
where 𝐶 and 𝑎 are in general complex numbers
Real exponential signals:
𝑥(𝑡) = 𝐶𝑒𝑎𝑡
, (𝑎) 𝑎 > 0; (𝑏) 𝑎 < 0
▪ Example:
2𝑒−2𝑡
, 0.4𝑒2𝑡

Periodic complex exponential and sinusoidal signals
If 𝑎 is purely imaginary, we have
𝑥(𝑡) = 𝑒𝑗𝜔0𝑡
▪ An important property of this signal is that it is periodic.
▪ The signals 𝑒𝑗𝜔0𝑡 and 𝑒−𝑗𝜔0𝑡 have the same fundamental period.
▪ Using Euler’s relation, a complex exponential can be expressed in terms of sinusoidal
signals with the same fundamental period:
𝑒𝑗𝜔0𝑡 = cos 𝜔0𝑡 + 𝑗 sin 𝜔𝑜𝑡

▪ A sinusoidal signal can also be expressed in terms of periodic complex
exponentials with the same fundamental period:
𝐴 cos(𝜔0𝑡 + 𝜙) =
𝐴
2
𝑒𝑗𝜙
𝑒𝑗𝜔0𝑡
+
𝐴
2
𝑒−𝑗𝜙
𝑒−𝑗𝜔0𝑡
▪ A sinusoid can also be expresses as
𝐴 cos(𝜔0𝑡 + 𝜙) = 𝐴 Re{𝑒𝑗(𝜔0𝑡+𝜙)}
and
𝐴 sin (𝜔0𝑡 + 𝜙) = 𝐴 Im{𝑒𝑗(𝜔0𝑡+𝜙)}

▪ Periodic signals, such as the sinusoidal signals provide important
examples of signal with infinite total energy, but finite average
power. For example:
𝐸𝑝𝑒𝑟𝑖𝑜𝑑 = ‫׬‬
0
𝑇0
𝑒𝑗𝜔0𝑡 2
𝑑𝑡 = ‫׬‬
0
𝑇0
1𝑑𝑡 = 𝑇0 and
𝑃𝑝𝑒𝑟𝑖𝑜𝑑 =
1
𝑇0
‫׬‬
0
𝑇0
𝑒𝑗𝜔0𝑡 2
𝑑𝑡 =
1
𝑇0
‫׬‬
0
𝑇0
1𝑑𝑡 = 1
▪ Since there are an infinite number of periods as t ranges from −∞ to
+ ∞, the total energy integrated over all time is infinite.
𝑃∞ = lim
𝑇→∞
1
2𝑇
‫׬‬
−𝑇
𝑇
𝑒𝑗𝜔0𝑡 2
𝑑𝑡 = 1

Harmonically related complex exponentials:
𝜙𝑘 𝑡 = 𝑒𝑗𝑘𝜔0𝑡 , 𝑘 = 0, ±1, ±2, …
𝜔0 is the fundamental frequency
▪ Example:
Signal 𝑥(𝑡) = 𝑒𝑗2𝑡 + 𝑒𝑗3𝑡

General complex Exponential signals:
Consider a complex exponential 𝐶𝑒𝑎𝑡
, where 𝐶 = 𝐶 𝑒𝑗𝜃
is expressed in polar and
𝑎 = 𝑟 + 𝑗𝜔0 is expressed in rectangular form. Then
𝐶𝑒𝑎𝑡
= 𝐶 𝑒𝑗𝜃
𝑒 𝑟+𝑗𝜔0 𝑡
= 𝐶 𝑒𝑟𝑡
𝑒𝑗(𝜔0𝑡+𝜃)
= 𝐶 𝑒𝑟𝑡 cos(𝜔0𝑡 + 𝜃) + 𝑗 𝐶 𝑒𝑟𝑡 sin(𝜔0𝑡 + 𝜃)
▪ Thus, for 𝑟 = 0 , the real and imaginary parts of a complex exponential are
sinusoidal.
▪ For 𝑟 > 0 , sinusoidal signals multiplied by a growing exponential.
▪ For 𝑟 < 0 , sinusoidal signals multiplied by a decaying exponential or Damped
signal.
a
(a) Growing sinusoidal signal; (b) decaying sinusoidal signal.

1.3.1 Discrete-time complex exponential and sinusoidal signals
A discrete-time complex exponential signal is defined by 𝑥[𝑛] = 𝐶𝛼𝑛
where 𝐶 and 𝛼 are in general complex numbers. Alternatively 𝑥[𝑛] = 𝐶𝑒𝛽𝑛
in which 𝛼 = 𝑒𝛽
.
Real exponential signals:
If 𝐶 and 𝛼 are real, we have the real
exponential signals.
Example:
𝑥 𝑛 = 𝐶𝛼𝑛:
a 𝛼 > 1; b 0 < 𝛼 < 1;
c − 1 < 𝛼 < 0; d 𝛼 < −1

Sinusoidal Signals:
If 𝑎 is purely imaginary, we have
𝑥[𝑛] = 𝑒𝑗𝜔0𝑛
𝑒𝑗𝑤0𝑛 = cos 𝜔0𝑛 + 𝑗 sin 𝜔0𝑛
▪ An important property of this signal is that it is periodic.
▪ The signals 𝑒𝑗𝑤0𝑛
and 𝑒−𝑗𝑤0𝑛
have the same fundamental period.
▪ Using Euler’s relation, a complex exponential can be expressed in
terms of sinusoidal signals with the same fundamental period:
𝑒𝑗𝑤0𝑛 = cos 𝜔0𝑛 + 𝑗 sin 𝜔𝑜𝑛

▪ A sinusoidal signal can also be expressed in terms of periodic complex
exponentials with the same fundamental period:
𝐴 cos(𝜔0𝑛 + 𝜙) =
𝐴
2
𝑒𝑗𝜙𝑒𝑗𝜔0𝑛 +
𝐴
2
𝑒−𝑗𝜙𝑒−𝑗𝜔0𝑛
▪ A sinusoid can also be expresses as
𝐴 cos(𝜔0𝑛 + 𝜙) = 𝐴 Re{𝑒𝑗(𝜔0𝑛+𝜙)}
and
𝐴 sin 𝜔0𝑛 + 𝜙) = 𝐴 Im{𝑒𝑗(𝜔0𝑛+𝜙)
}
Example:

General complex Exponential signals:
Consider a complex exponential 𝐶𝛼𝑛
, where 𝐶 = 𝐶 𝑒𝑗𝜃
and 𝛼 = 𝛼 𝑒𝑗𝜔0 then,
𝐶𝛼𝑛
= 𝐶 𝛼 𝑛
cos 𝜔0𝑛 + 𝜃 + 𝑗 𝐶 𝛼 𝑛
cos 𝜔0𝑛 + 𝜃
▪ Thus, for 𝛼 = 1 , the real and imaginary parts of a complex exponential are
sinusoidal.
▪ For 𝛼 > 1 , sinusoidal signals multiplied by a growing exponential.
▪ For 𝛼 < 1 , sinusoidal signals multiplied by a decaying exponential or
Damped signal.
a
(a) Growing sinusoidal signal; (b) decaying sinusoidal signal.

1.3.3 Periodicity Properties of Discrete-Time Complex Exponentials
There are a number of important differences between CT and DT sinusoidal
signals.
▪ CT signals 𝑒𝑗𝜔0𝑡 are distinct for distinct values of 𝜔0.
▪ DT signals 𝑒𝑗𝜔0𝑛 are not distinct for all distinct values of 𝜔0. 𝑒𝑗𝜔0𝑛 is
identical to the signals with frequencies 𝜔0 ± 2𝜋, 𝜔0 ± 4𝜋, …
𝑒𝑗𝜔0𝑛 = 𝑒𝑗(𝜔0±2𝜋)𝑛 = 𝑒𝑗(𝜔0±4𝜋)𝑛

Example: To understand why DT signals have identical values for 𝜔0
and 𝜔0 ± 2𝑘𝜋, consider the two sinusoidal signals sin(
𝜋
4
𝑡) and sin(
𝜋
4
+ 2𝜋)𝑡:

▪ For 𝑒𝑗𝜔0𝑛
, when 𝜔0 is increased (from zero), the oscillation rate increases
until 𝜔0 reaches 𝜋. When 𝜔0 is continuously increased (after 𝜋), the
oscillation rate decreases until 𝜔0 reaches 2𝜋.
▪ Conclusion: the low-frequency discrete-time exponentials have values of
𝜔0 near 0, 2𝜋, 4𝑘𝜋, …, while the high-frequencies are located near 𝜔0 =
𝜋, (2𝑘 + 1)𝜋.

Fundamental frequency and fundamental period for 𝒙 𝒏 = 𝒆𝒋𝝎𝟎𝒏
:
A DT signal is periodic if 𝑥 𝑛 = 𝑥[𝑛 + 𝑁]
𝑒𝑗𝜔0𝑛
= 𝑒𝑗𝜔0(𝑛+𝑁)
֜ 𝑒𝑗𝜔0𝑁
= 1 ֜ 𝜔0𝑁 = 2𝜋𝑚 𝑚 = 0, ±1, ±2, …
֜
𝜔0
2𝜋
=
𝑚
𝑁
Conclusion: 𝑥 𝑛 = 𝑒𝑗𝜔0𝑛
is a periodic signal if
𝝎𝟎
𝟐𝝅
is a rational number.
Fundamental frequency:
2𝜋
𝑁
=
𝜔0
𝑚
Fundamental period: 𝑁 = 𝑚
2𝜋
𝜔0
, when 𝑁 is an integer
Fundamental frequency

Example: Determine the fundamental period of the following DT signal:
𝑥 𝑛 = 𝑒𝑗(2𝜋/3)𝑛 + 𝑒𝑗(3𝜋/4)𝑛

Harmonically related periodic exponentials:
Consider a periodic exponentials signal like 𝑒𝑗𝜔0𝑛 with fundamental period
2𝜋
𝑁
=
𝜔0
𝑚
.
𝜙𝑘[𝑛] = 𝑒
𝑗𝑘
2𝜋
𝑁 𝑛
, 𝑘 = 0, ±1, ±2, …
For DT signals, there are only N distinct period exponentials in the set 𝜙𝑘[𝑛].
This can be proved as follows:
𝜙𝑘+𝑁 𝑛 = 𝑒
𝑗 𝑘+𝑁
2𝜋
𝑁
𝑛
= 𝑒
𝑗2
𝑘𝜋
𝑁
𝑛
𝑒𝑗2𝜋𝑛 = 𝑒
𝑗2
𝑘𝜋
𝑁
𝑛
= 𝜙𝑘[𝑛]

1.4 The Unit Impulse and Unit Step Functions
Discrete-Time Unit Impulse and Unit Step Sequences:
Discrete-time unit impulse is defined as
𝛿[𝑛] = ቊ
0, 𝑛 ≠ 0
1, 𝑛 = 0
Discrete-time unit step is defined as
𝑢[𝑛] = ቊ
0, 𝑛 < 0
1, 𝑛 ≥ 0
0
1

The discrete-time unit impulse is the first difference of the discrete-time step
𝛿 𝑛 = 𝑢 𝑛 − 𝑢[𝑛 − 1]
The discrete-time unit step is the running sum of the unit sample:
𝑢 𝑛 = ෍
𝑚=−∞
𝑛
𝛿[𝑚]
or
𝑢 𝑛 = ෍
𝑘=0
∞
𝛿[𝑛 − 𝑘]
or
𝑢 𝑛 = ෍
𝑘=−∞
∞
𝑢[𝑘]𝛿[𝑛 − 𝑘]

▪ Sampling Property of 𝜹 𝒏 :
𝑥 𝑛 𝛿 𝑛 = 𝑥[0]𝛿 𝑛
𝑥[𝑛]𝛿[𝑛 − 𝑛0] = ቊ
𝑥[𝑛], 𝑛 = 𝑛0
0, 𝑛 ≠ 𝑛0
𝑥 𝑛 𝛿 𝑛 − 𝑛0 = 𝑥[𝑛0]𝛿 𝑛 − 𝑛0

▪ Representation Property of 𝜹 𝒏 :
𝑥[𝑘]𝛿[𝑛 − 𝑘] = 𝑥[𝑛]𝛿[𝑛 − 𝑘]
Summing the both sides over the index k yields
෍
𝑘=−∞
∞
𝑥[𝑘]𝛿[𝑛 − 𝑘] = ෍
𝑘=−∞
∞
𝑥[𝑛]𝛿[𝑛 − 𝑘] = 𝑥[𝑛] ෍
𝑘=−∞
∞
𝛿 𝑛 − 𝑘 = 𝑥[𝑛]
▪ This result shows that every discrete-time signal x[n] can be represented as a
linear combination of shifted unit impulses.
𝑥 𝑛 = ෍
𝑘=−∞
∞
𝑥[𝑘]𝛿[𝑛 − 𝑘]
Example:
𝑢 𝑛 = ෍
𝑘=−∞
∞
𝑢[𝑘]𝛿[𝑛 − 𝑘]

▪ Representation Property of 𝜹 𝒏 :
Example: Representing of a signal 𝑥[𝑛] using a train of impulses 𝛿[𝑛 − 𝑘]

The Continuous-Time Unit Step and Unit Impulse Functions:
Continuous-time unit step is defined as
𝑢(𝑡) = ቊ
0, 𝑡 < 0
1, 𝑡 ≥ 0
, 𝑢 𝑡 = ‫׬‬
−∞
𝑡
𝛿(𝜏)𝑑𝜏,
𝛿Δ 𝑡 =
𝑑𝑢Δ(𝑡)
𝑑𝑡
𝛿Δ(𝑡) = ൝
1
Δ
, 0 ≤ 𝑡 < Δ
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝛿 𝑡 = lim
Δ→0
𝛿Δ(𝑡)
𝛿 𝑡 =
𝑑𝑢(𝑡)
𝑑𝑡

Example:

▪ Sampling property:
𝑥 𝑡 𝛿 𝑡 = 𝑥(0)𝛿 𝑡 or more generally
𝑥 𝑡 𝛿 𝑡 − 𝑡0 = 𝑥(𝑡0)𝛿 𝑡 − 𝑡0
𝑢(𝑡) = ቊ
0, 𝑡 < 0
1, 𝑡 ≥ 0
, 𝑢 𝑡 = ‫׬‬
−∞
𝑡
𝛿(𝜏)𝑑𝜏,
▪ Shifting Property:
The shifting property follows from the sampling property. Integrating
𝑥 𝑡 𝛿(𝑡) yields
න
−∞
∞
𝑥 𝑡 𝛿 𝑡 𝑑𝑡 = න
−∞
∞
𝑥 0 𝛿 𝑡 𝑑𝑡 = 𝑥 0 න
−∞
∞
𝛿 𝑡 𝑑𝑡 = 𝑥 0
න
−∞
∞
𝑥 𝑡 𝛿 𝑡 − 𝑡0 𝑑𝑡 = 𝑥(𝑡0)

▪ Representation Property:
𝑥 𝑡 = න
−∞
∞
𝑥 𝜏 𝛿 𝑡 − 𝜏 𝑑𝜏

1.5 Continuous-Time and Discrete-Time Systems
▪ System: A system is a quantitative description of a physical process which
transforms signals (at its “input”) to signals (at its “output”).
Example (CT System):
▪ force f (t) as the input and the velocity 𝑣(𝑡) as the output
▪ m denote the mass of the car and 𝜌𝑣(𝑡) the resistance due to friction
𝑑𝑣(𝑡)
𝑑𝑡
=
1
𝑚
[𝑓 𝑡 − 𝜌𝑣 𝑡 ]
First-order linear differential equation:
𝑑𝑦(𝑡)
𝑑𝑡
+ 𝑎𝑦(𝑡) = 𝑏𝑥(𝑡)

Example (DT System):
▪ System is the balance in a bank account from month to month.
▪ 𝑦[𝑛] denotes the balance at the end of 𝑛𝑡ℎ month, and 𝑥[𝑛], is the net deposit
(deposits minus withdraws).
▪ The account has 1% interest each month
𝑦 𝑛 = 1.01𝑦 𝑛 − 1 + 𝑥[𝑛]
First-order linear differential equation (DT):
𝑦[𝑛] + 𝑎𝑦[𝑛 − 1] = 𝑏𝑥[𝑛]

Interconnections of Systems:
(a) Series or cascade interconnection
(b) Parallel interconnection
(c) Combination of both series and parallel

Feedback interconnection

1.6 Basic System Properties
Systems with and without Memory
A system is memoryless if its output for each value of the independent variable
as a given time is dependent only on the input at the same time.
Example 1:
𝑦[𝑛] = 2𝑥 𝑛 − 𝑥2
𝑛 2
Example 2: Accumulator or summer
𝑦 𝑛 = ෍
𝑘
𝑛
𝑥 𝑘
Example 3: Delay
𝑦 𝑛 = 𝑥[𝑛 − 1]
Example 4: A capacitor
𝑣 𝑡 =
1
𝐶
‫׬‬
−∞
𝑡
𝑖 𝜏 𝑑𝜏

Causality
▪ A system is causal (non-anticipative) if the output at any time depends
only on the values of the input at present time and in the past.
▪ All memoryless systems are causal since the output responds only to the
current value of input.
Example1 : 𝑅𝐶 Circuit is a causal system
Example 2: The following system is not causal
𝑦 𝑛 = 𝑥 𝑛 − 𝑥[𝑛 + 1]
▪ Example 3: Determine the Causality of the two systems:
(1) 𝑦[𝑛] = 𝑥[−𝑛]
(2) 𝑦(𝑡) = 𝑥(𝑡)cos(𝑡 + 1)

Time Invariance
A system is time invariant if a time shift in the input signal results in an
identical time shift in the output signal.
▪ Mathematically, if the system output is 𝑦(𝑡) when the input is 𝑥(𝑡) , a time-
invariant system will have an output of 𝑦(𝑡 − 𝑡0) when input is 𝑥(𝑡 − 𝑡0).
Example: 𝑦(𝑡) = sin[𝑥(𝑡)]
▪ The use of counter-example is the best way to show a system is not time
invariant.
Example: 𝑦[𝑛] = 𝑛𝑥[𝑛]

Example: 𝑦 𝑡 = 𝑥(2𝑡)
𝑦1 𝑡 − 2 = 𝑥1(2(𝑡 − 2))
≠
𝑦2 𝑡 = 𝑥2 2𝑡 = 𝑥1(2𝑡 − 2)
𝑦1 𝑡 = 𝑥1 2𝑡

Linearity
A system is linear if
▪ The response to 𝑥1 𝑡 + 𝑥2(𝑡) is 𝑦1 𝑡 + 𝑦2(𝑡) - additivity property
▪ The response to 𝛼𝑥1 𝑡 is 𝛼𝑦1 𝑡 - scaling or homogeneity property.
or
𝛼𝑥1 𝑡 + 𝛽𝑥2 𝑡 → 𝛼𝑦1 𝑡 + 𝛽𝑦2(𝑡) CT systems
𝛼𝑥1 𝑛 + 𝛽𝑥2 𝑛 → 𝛼𝑦1[𝑛] + 𝛽𝑦2[𝑛] DT systems
Superposition property:
If 𝑥𝑘 𝑛 , 𝑘 = 1,2,3, … are a set of inputs with corresponding outputs
𝑦𝑘 𝑛 , 𝑘 = 1,2,3, …, then the response to a linear combination of these inputs
given by
𝑥 𝑛 = 𝑎1𝑥1 𝑛 + 𝑎2𝑥2 𝑛 + 𝑎3𝑥3 𝑛 + ⋯
is
𝑦 𝑛 = 𝑎1𝑦1 𝑛 + 𝑎2𝑦2 𝑛 + 𝑎3𝑦3 𝑛 + ⋯

Example:
(1) 𝑦 𝑡 = 𝑡𝑥(𝑡) is a linear system
(2) 𝑦 𝑡 = 𝑥2
(𝑡) is not a linear system
(3) 𝑦[𝑛] = 2𝑥 𝑛 + 3 is not a linear system
(4) 𝑦[𝑛] = 2𝑥[𝑛] is a linear system
(5) 𝑦[𝑛] = 𝑅𝑒{𝑥[𝑛]} is not a linear system

Summary
▪ We learn the notion of Signal Energy and Power
▪ We learn the techniques of “Transformations of the independent variable”
▪ We learn DT and CT signals and some of their properties particularly for
periodic DT and CT signals
▪ Some of fundamental DT and CT signals including Unit Impulse and Unit
Step signals were reviewed.
▪ CT and DT systems and some of their properties, including Memoryless,
Causality, Time Invariance and Linearity were introduced.

MSE280s Chapter1_Signals_and_Systems.pdf

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