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Signals & Systems

Lecture 2 – Signals and Sinusoidals

• Signal: Pattern of variations of a physical quantity that carries information and that can be manipulated, stored or transmitted by physical processes

• System: Something that can manipulate, store or transmit signals

Signals & Systems

• Signal: A function of one or more variables that conveys information on the nature of a physical phenomenon

• System: An entity that manipulates one or more signals to accomplish a function, thereby yielding new signals

Signals & Systems

• Signal: Function of one or more independent variable, contains information about the behavior or nature of some phenomenon

• System: Responds to a signal particular signal by producing other signals or some desired behavior

Signals & Systems

• The information in a signal is contained in a pattern of variations of some form

• Signals are represented mathematically as functions of one or more independent variables

• For example, a speech signal can be represented by acoustic pressure as a function of time

• As another example, a picture can be represented by brightness as a function of two spatial variables

Continuous-Time / Discrete-Time

• Continuous-Time Signals: The independent variable is continuous. Hence, these signals are defined for a continuum of values of the independent variable

• Discrete-Time Signals: The independent variable takes on only a discrete set of values. Hence, these signals signals are defined only at discrete times.

• Discrete-time signals are also often derived from continuous-time signals by sampling at at uniform rate


• Continuous-time signal examples:

• Mass and spring

• Tanks and water

• Speech signal

• Atmospheric pressure as a function of altitude


• Discrete-time signal examples:

• Bank accounts

• Weekly stock prices

• Daily average temperatures

• Monthly unemployement ratios

• Image signal


• Continuous-time signal notation: x(t)

• Discrete-time signal notation: x[n]

• Discrete-time signal notation: ... x[-2] , x [-1] , x[0] , x[1] , x[2] , ...


• In many, although not all, applications, the signals are related to physical qualities quantities capturing power and energy

• If voltage and current across a resistor R are v(t) and i(t) respectively, then the instantaneous power is:

𝑝 𝑡 = 𝑣 𝑡 𝑖 𝑡 =1

𝑅𝑣2(𝑡)

Signal Energy & Power

• The total energy expanded over the time interval t1≤t ≤t2:

𝑡1

𝑡2

𝑝 𝑡 𝑑𝑡 = 𝑡1

𝑡2 1

𝑅𝑣2(𝑡) 𝑑𝑡

• The average energy expanded over the time interval t1≤t ≤t2:

1

𝑡2 − 𝑡1 𝑡1

𝑡2

𝑝 𝑡 𝑑𝑡 =1

𝑡2 − 𝑡1 𝑡1

𝑡2 1

𝑅𝑣2(𝑡) 𝑑𝑡


• For complex continuous-time signals:

𝑡1

𝑡2

𝑥(𝑡) 2𝑑𝑡

• For complex discrete-time signals:

𝑛1

𝑛2

𝑥[𝑛] 2


• Total energy for complex continuous-time signals, for infinite time:

𝐸∞ = lim𝑇→∞

−𝑇

𝑇

𝑥(𝑡) 2𝑑𝑡

• Total energy for complex discrete-time signals, for infinite time:

𝐸∞ = lim𝑁→∞

𝑛1

N

𝑥[𝑛] 2


• A central concept in signals & systems

• We will focus on an elementary transformation: time shift

• These elementary transformation allow us to introduce several basic properties of signals & systems

• Later on, they will also play part in defining and characterizing far richer and important classes of systems

Transformation of the Independent Variable

• Time shift:


• Time shift:

• Two signals x[n] and x[𝑛 − 𝑛0] that are identical in shape, but are displaced or shifted with respect to each other

• x[𝑛 − 𝑛0] is a delayed, x[𝑛 + 𝑛0] is an advanced signal

• Typical in radar, sonar, and seismic signal processing (several receivers at different locations)


• Time reversal:


• Time scaling:


• Overall Example: What is the relation between the signals given below?


• Overall Example:


• A periodic continuous-time signal x(t) has the propert that there is a positive value of T for which

𝑥 𝑡 = 𝑥(𝑡 + 𝑇)

for all values of t.

• In other words, a periodic signal has the property that it is unchanged by a time shift of T. In this case, we say that x(t) is periodic with a period of T.

Periodic Signals

• Given above is a periodic signal x(t). We can deduce that

x(t) = x(t + mT)

for all t and for any integer m.

• x(t) is also periodic with period 2T, 3T, 4T, ... The fundamental period 𝑇0 is the smallest positive value of T for which the equation above holds.

Periodic Signals

• Fundamental frequency is the reciprocal of fundamental period

𝑓 =1

𝑇

• The frequency f is measured in hertz (Hz) or cycles per second.

• The angular frequency is measured in radians per second and is defined by:

𝜔 =2𝜋

𝑇

Periodic Signals

Periodic Signals

Periodic Signals

• What is the fundamental frequency of the wave below? Express in Hz and rad/s.

• 5Hz or 10 π rad/s

Periodic Signals

• What is the fundamental frequency of the wave below in rad/s?

• π/4 rad/s

• A dicrete-time signal is even if x[-n] = x[n]

• A continuous-time signal is odd if x(-t) = - x(t)

• A dicrete-time signal is odd if x[-n] = - x[n]

• Note that an odd signal must necessarily be 0 at t = 0 or n = 0

Even and Odd Signals

• A continuous-time signal is even if x(-t) = x(t)

𝑂𝑑𝑑 𝑥 𝑡 =1

2𝑥 𝑡 − 𝑥 −𝑡


• An important fact is that any signal can be broken into a sum of two signals, one of which is even, and one of which is odd.

𝐸𝑣𝑒𝑛 𝑥 𝑡 =1

2𝑥 𝑡 + 𝑥 −𝑡

• The continuous-time complex exponential signal is of the form:

𝑥 𝑡 = 𝐶𝑒𝑎𝑡

• Examples include:

• Atomic explosions and chemical reactions for a > 0

• Radioactive decay and the responses of RC circuits for a < 0

Exponential Signals

Exponential Signals

• Consider a to be purely imaginary:

𝑥 𝑡 = 𝑒𝑗𝜔0𝑡

• This signal is periodic: 𝑒𝑗𝜔0𝑡 = 𝑒𝑗𝜔0 𝑡+𝑇

𝑒𝑗𝜔0𝑇 = 1 ⇒ 𝑇0 =2𝜋

𝜔0

Exponential & Sinusoidal Signals

• A signal closely related to the periodic complext exponential is the sinuosidal signal

𝑥 𝑡 = 𝐴 cos 𝜔0𝑡 + 𝜙

Sinusoidal Signals

Sinusoidal Signals

𝑒𝑗𝜔0𝑡 = cos 𝜔0𝑡 + 𝑗 sin 𝜔0𝑡

𝐴 cos 𝜔0𝑡 + 𝜙 =𝐴2𝑒𝑗𝜙𝑒𝑗𝜔0𝑡 +

𝐴2𝑒−𝑗𝜙𝑒𝑗𝜔0𝑡

Sinusoidal Signals


Sinusoidal Signals

• A is called Amplitude. It is scaling factor that determines how large the sinusoidal signal will be.

• 𝜙 is called Phase Shift. Its units is radians.

• 𝜔0 is called the radian frequency. If t has units of seconds, 𝜔0has rad/s as units.

𝑥 𝑡 = 5 cos 0.3𝜋𝑡 + 1.2𝜋

• A = 5 , w = 0.3 π , φ = 1.2 π

• T = 2 π / w = 20 /3

Sinusoidal Signals

Sinusoidal Signals

Sinusoidal Signals: Damped / Growing• A damped or growing sinusoid is given by

𝑥 𝑡 = 𝑒𝜎𝑡 cos 𝜔0𝑡 + 𝜙

• Exponential growth if 𝜎 > 0 , decay if 𝜎 < 0


Sinusoidal Signals: Phase Shift

• The phase shift parameter 𝜙 determines the locations of the maxima and minima of the cosine wave.

• The cosine with 𝜙 = 0 has a positive peak at t = 0.

• When 𝜙 ≠ 0, the phase shift determines how much the maximum of the cosine signal is shifted away from t = 0.

𝑥 𝑡 = 5 cos 0.3𝜋𝑡 + 1.2𝜋 = 5 cos 0.3𝜋 𝑡 − −4



• For a signal given by:

𝑥 𝑡 = 𝐴 cos 𝜔0𝑡

• Applying a time shift:

𝑥 𝑡 − 𝑡1 = 𝐴 cos 𝜔0 𝑡 − 𝑡1 =𝐴 cos 𝜔0𝑡 − 𝜔0𝑡1

𝜙 = −𝜔0𝑡1 = −2𝜋𝑓0𝑡1 = −2𝜋𝑡1𝑇0


• The positive peak nearest to t = 0 must always lie within

𝑡1 ≤ 𝑇0 2

• Therefore phase shift can always be chosen to satisfy

−𝜋 ≤ 𝜙 < 𝜋

• However, we can also add a multiple of 2𝜋, and it does not change the value of the cosine

Sinusoidal Signals: Plotting


• Plotting the following signal in MATLAB

𝑥 𝑡 = 20 cos 2𝜋 40 𝑡 − 0.4𝜋

n = -10:10;

Ts = 0.005;

tn = n * Ts;

xn = 20 * cos ( 80 * pi * tn - 0.4 * pi );

figure; plot ( tn , xn );



𝑥 𝑡 = 20 cos 2𝜋 40 𝑡 − 0.4𝜋

n = -10:10;

Ts = 0.0025;

tn = n * Ts;

xn = 20 * cos ( 80 * pi * tn - 0.4 * pi );

figure; plot ( tn , xn );



𝑥 𝑡 = 20 cos 2𝜋 40 𝑡 − 0.4𝜋

n = -10:10;

Ts = 0.0025;

tn = n * Ts;

xn = 20 * cos ( 80 * pi * tn - 0.4 * pi );

figure; stem ( tn , xn );



𝑥 𝑡 = 20 cos 2𝜋 40 𝑡 − 0.4𝜋

n = -10:10;

Ts = 0.0001;

tn = n * Ts;

xn = 20 * cos ( 80 * pi * tn - 0.4 * pi );




𝑥 𝑡 = 20 cos 2𝜋 40 𝑡 − 0.4𝜋

n = -100:100;

Ts = 0.0001;

tn = n * Ts;

xn = 20 * cos ( 80 * pi * tn - 0.4 * pi );




𝑥 𝑡 = 20 cos 2𝜋 40 𝑡 − 0.8𝜋

n = -100:100;

Ts = 0.0001;

tn = n * Ts;

xn = 20 * cos ( 80 * pi * tn - 0.8* pi );


Variable Transformation Examples

• MATLAB Audio Example


• Which images below match the expressions beneath them?

Sinusoidal Signals:Phasor Addition

• We occasionally add several sinusoidal signals with the same frequency, but with different amplitudes and phases

• The following statement is satisfied in this case:

𝑘=1

𝑁

𝐴𝑘 cos 𝜔0𝑡 + 𝜙𝑘 = 𝐵 cos 𝜔0𝑡 + 𝜃

• In other words, we obtain a single sinusoidal signal with the same frequency.


• An important trick is to use the following trigonometric identities:

𝐴𝑘 cos 𝜔0𝑡 + 𝜙𝑘 = 𝐴𝑘 cos 𝜙𝑘 cos 𝜔0𝑡 − 𝐴𝑘 sin 𝜙𝑘 sin 𝜔0𝑡

𝑘=1

𝑁

𝐴𝑘 cos 𝜔0𝑡 + 𝜙𝑘 =

𝑘=1

𝑁

𝑅𝑒 𝐴𝑘𝑒𝑗 𝜔0𝑡+𝜙𝑘 =

𝑘=1

𝑁

𝑅𝑒 𝐴𝑘𝑒𝑗𝜔0𝑡𝑒𝑗𝜙𝑘

= 𝑅𝑒

𝑘=1

𝑁

𝐴𝑘𝑒𝑗𝜙𝑘 𝑒𝑗𝜔0𝑡


1.7 cos 20𝜋𝑡 + 70𝜋/180 + 1.9 cos 20𝜋𝑡 + 200𝜋/180 = ?


1.7 cos 20𝜋𝑡 + 70𝜋/180 + 1.9 cos 20𝜋𝑡 + 200𝜋/180 = ?

• Represent the signal by phasors:

𝑋1 = 𝐴1𝑒𝑗𝜃1 = 1.7𝑒𝑗70𝜋/180

𝑋2 = 𝐴2𝑒𝑗𝜃2 = 1.9𝑒𝑗200𝜋/180

• Convert into rectangular form:

𝑋1 = 1.7 cos 70𝜋/180 + 𝑗1.7 sin 70𝜋/180𝑋2 = 1.9 cos 200𝜋/180 + 𝑗1.9 sin 200𝜋/180


1.7 cos 20𝜋𝑡 + 70𝜋/180 + 1.9 cos 20𝜋𝑡 + 200𝜋/180 = ?

• Represent the signal by phasors:

𝑋1 = 𝐴1𝑒𝑗𝜃1 = 1.7𝑒𝑗70𝜋/180

𝑋2 = 𝐴2𝑒𝑗𝜃2 = 1.9𝑒𝑗200𝜋/180

• Convert into rectangular form:

𝑋1 = 0.5814 + 𝑗1.597𝑋2 = −1.785 − 𝑗0.6498


1.7 cos 20𝜋𝑡 + 70𝜋/180 + 1.9 cos 20𝜋𝑡 + 200𝜋/180 = ?

• Add the real and imaginary parts:

𝑋3 = 𝑋1 + 𝑋2 = 0.5814 + 𝑗1.597 + −1.785 − 𝑗0.6498

𝑋3 = −1.204 + 𝑗0.9476

• Convert back to polar form:

𝑋3 = −1.204 2 + 0.9476 2𝑒𝑗 tan−1 0.9476

−1.204


1.7 cos 20𝜋𝑡 + 70𝜋/180 + 1.9 cos 20𝜋𝑡 + 200𝜋/180 = ?

• Add the real and imaginary parts:

𝑋3 = 𝑋1 + 𝑋2 = 0.5814 + 𝑗1.597 + −1.785 − 𝑗0.6498

𝑋3 = −1.204 + 𝑗0.9476

• Convert back to polar form:

𝑋3 = 1.532𝑒𝑗141.79𝜋/180


1.7 cos 20𝜋𝑡 + 70𝜋/180 + 1.9 cos 20𝜋𝑡 + 200𝜋/180 = ?

= 1.532 cos 20𝜋𝑡 + 141.79𝜋/180

Unit Impulse and Unit Step• Unit impulse signal is defined as:

𝛿 𝑛 = 0 , 𝑛 ≠ 01 , 𝑛 = 0

Unit Impulse and Unit Step• Unit step signal is defined as:

𝑢 𝑛 = 0 , 𝑛 < 01 , 𝑛 ≥ 0

Unit Impulse and Unit Step• Unit impulse signal can also be represented in terms of unit

step signals as:

𝛿 𝑛 = 𝑢 𝑛 − 𝑢[𝑛 − 1]

• Unit step signal can also be represented in terms of unit impulse signals as:

𝑢 𝑛 =

𝑚=−∞

𝑛

𝛿[𝑚]

Unit Impulse and Unit Step• Unit step signal can also be represented in terms of unit

impulse signals as:

𝑢 𝑛 =

𝑚=−∞

𝑛

𝛿[𝑚]


impulse signals as:

𝑢 𝑛 =

𝑚=−∞

𝑛

𝛿[𝑚]

𝑢 𝑛 =

𝑘=∞

0

𝛿[𝑛 − 𝑘]

𝑢 𝑛 =

𝑘=0

∞

𝛿[𝑛 − 𝑘]


impulse signals as:

𝑢 𝑛 =

𝑘=0

∞

𝛿[𝑛 − 𝑘]

Unit Impulse and Unit Step

• Unit impulse signal can be used to find the value of a signal at 𝑛 = 0:

𝑥 𝑛 𝛿 𝑛 = 𝑥[0]𝛿 𝑛

• Or more generally 𝑛 = 𝑛0 by:

𝑥 𝑛 𝛿 𝑛 − 𝑛0 = 𝑥[𝑛0]𝛿 𝑛 − 𝑛0

Unit Impulse and Unit Step• Unit step signal in continuous-time is defined as:

𝑢(𝑡) = 0 , 𝑡 < 01 , 𝑡 > 0

Unit Impulse and Unit Step• Continuous-time unit step and continuous-time unit impulse

signals are related by:

𝑢 𝑡 = −∞

𝑡

𝛿 𝜏 𝑑𝜏 = 0

∞

𝛿 𝑡 − 𝜎 𝑑𝜎

Unit Impulse and Unit Step• Continuous-time unit step and continuous-time unit impulse

signals are related by:

𝛿(𝑡) =𝑑𝑢(𝑡)

𝑑𝑡

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