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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 / Discrete-Time
Continuous-Time / Discrete-Time
• Continuous-time signal examples:
• Mass and spring
• Tanks and water
• Speech signal
• Atmospheric pressure as a function of altitude
Continuous-Time / Discrete-Time
• Discrete-time signal examples:
• Bank accounts
• Weekly stock prices
• Daily average temperatures
• Monthly unemployement ratios
• Image signal
Continuous-Time / Discrete-Time
• 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] , ...
Continuous-Time / Discrete-Time
• 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
Signal Energy & Power
• For complex continuous-time signals:
𝑡1
𝑡2
𝑥(𝑡) 2𝑑𝑡
• For complex discrete-time signals:
𝑛1
𝑛2
𝑥[𝑛] 2
Signal Energy & Power
• 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
Signal Energy & Power
• 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:
Transformation of the Independent Variable
• Time shift:
Transformation of the Independent Variable
• 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)
Transformation of the Independent Variable
• Time reversal:
Transformation of the Independent Variable
• Time reversal:
Transformation of the Independent Variable
• Time scaling:
Transformation of the Independent Variable
• Time scaling:
Transformation of the Independent Variable
• Overall Example: What is the relation between the signals given below?
Transformation of the Independent Variable
Transformation of the Independent Variable
• Overall Example:
Transformation of the Independent Variable
• 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
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)
Even and Odd Signals
Even and Odd Signals
𝑂𝑑𝑑 𝑥 𝑡 =1
2𝑥 𝑡 − 𝑥 −𝑡
Even and Odd Signals
• 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
Even and Odd Signals
• 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
Sinusoidal Signals
𝑒𝑗𝜔0𝑡 = cos 𝜔0𝑡 + 𝑗 sin 𝜔0𝑡
𝐴 cos 𝜔0𝑡 + 𝜙 =𝐴2𝑒𝑗𝜙𝑒𝑗𝜔0𝑡 +
𝐴2𝑒−𝑗𝜙𝑒𝑗𝜔0𝑡
Sinusoidal Signals
𝑥 𝑡 = 𝐴 cos 𝜔0𝑡 + 𝜙
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
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
𝑥 𝑡 = 𝐴 cos 𝜔0𝑡 + 𝜙
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
Sinusoidal Signals: Phase Shift
Sinusoidal Signals: Phase Shift
• 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
Sinusoidal Signals: Phase Shift
• 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
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 );
Sinusoidal Signals: Plotting
• Plotting the following signal in MATLAB
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 );
Sinusoidal Signals: Plotting
• Plotting the following signal in MATLAB
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 );
Sinusoidal Signals: Plotting
• Plotting the following signal in MATLAB
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 );
figure; stem ( tn , xn );
Sinusoidal Signals: Plotting
• Plotting the following signal in MATLAB
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 );
figure; stem ( tn , xn );
Sinusoidal Signals: Plotting
• Plotting the following signal in MATLAB
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 );
figure; stem ( tn , xn );
Variable Transformation Examples
• MATLAB Audio Example
Variable Transformation Examples
• Which images below match the expressions beneath them?
Variable Transformation Examples
Variable Transformation Examples
• 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.
Sinusoidal Signals:Phasor Addition
• 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
Sinusoidal Signals:Phasor Addition
1.7 cos 20𝜋𝑡 + 70𝜋/180 + 1.9 cos 20𝜋𝑡 + 200𝜋/180 = ?
Sinusoidal Signals:Phasor Addition
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
Sinusoidal Signals:Phasor Addition
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
Sinusoidal Signals:Phasor Addition
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
Sinusoidal Signals:Phasor Addition
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
Sinusoidal Signals:Phasor Addition
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]
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]
𝑢 𝑛 =
𝑘=∞
0
𝛿[𝑛 − 𝑘]
𝑢 𝑛 =
𝑘=0
∞
𝛿[𝑛 − 𝑘]
Unit Impulse and Unit Step• Unit step signal can also be represented in terms of unit
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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