Signals and Sytems

Name: Signals and Sytems
Rating: 4.6 (35 reviews)

Learn fundamentals of signals and systems, by using helpful tools such as Fourier transform.

Created byAfterclap Team

Last updated 12/2022

English

What you'll learn

Being able to recognize commonly used signals like; the unit step, ramp, impulse function, sinusoidal signals and complex exponentials.
Describe signals mathematically and understand how to perform mathematical operations on signals properly.
Being able to apply Fourier Series and Transform with ease, learning the application with a deep understanding.
Applying Fourier transform and inverse Fourier transform to a wide range of different signals.
Be able to make easy classifications of signals such as ; even/odd, periodic/non-periodic, discrete/continuous time.
Learning the definition of the systems.
Learning the properties of signals and systems that help you to classify any system you encounter.
Basic convolution in continuous and discrete time domains.

Course content

4 sections • 47 lectures • 10h 35m total length

1.2.1 Examples and Mathematical Representation of Signals9:32
Present signals and their mathematical representation as functions. Distinguish continuous time signals from discrete time signals using ct s with brackets and t, and dts with square brackets and n.
1.2.2 Signal Energy and Power13:01
Compute energy and average power for continuous and discrete time signals using integrals and sums, and classify signals into finite energy, finite power, or neither.
1.3.1 Transformation of the Independent Variable17:05
Explore how transforming the independent variable alters signals—shift, reverse, and scale signals, with discrete-time examples like f(t-n), f(-t), and f(a t) in audio systems.
Example - 1 Transformation of the Independent Variable9:46
Explore how to transform the independent variable in a signal through shifting, reflection, and scaling of X(T). Understand positive versus negative variable rules, including X(T+1), X(-T+1), and X(3/2 T).
Quiz for Transformation of the Independent Variable
1.3.2 Periodic Signals5:59
Define periodic signals as those that repeat over a time interval, and identify the fundamental period as the smallest positive repetition; illustrate with continuous and discrete notations.
Quiz for Periodic Signals
Example - 2 Periodic Signals3:49
The lecture analyzes whether a piecewise x(t), cosine for t<0 and sine for t>=0, is periodic. It concludes the discontinuity at t=0 means x(t) is non-periodic.
1.3.3 Even and Odd Signals9:10
Examine even and odd signals, where even satisfies f(-t)=f(t) and odd satisfies f(-t)=-f(t). Learn to decompose any signal into even and odd parts with symmetry about the y-axis and origin.
1.4.1 Continuous Time Complex Exponential and Sinusoidal Signals14:18
Study continuous-time complex exponential and sinusoidal signals, including real and purely imaginary exponentials and their growth, decay, and periodicity. Explore Euler’s formula and the link between complex exponentials and sinusoids.
Example - 3 - Continuous Time Complex Exponential and Sinusoidal Signals7:58
Convert a complex exponential signal into a sinusoidal using Euler's formula, express cosine as (e^{jωt}+e^{-jωt})/2, and plot using the resulting cosine form.
1.4.2 General Complex Exponential Signals6:46
Explore general complex exponential signals in continuous time using c e^{alpha t} with alpha = r + j omega, and apply Euler's relation to see growth, decay, and damped sinusoids.
1.4.3 Discrete Time Complex Exponentials and Sinusoidal Signals13:42
Explore discrete-time complex exponentials and sinusoidal signals, including real, decaying, and growing exponentials, and learn how Euler’s relation links them to cosine and sine in discrete time.
1.4.4 Periodicity Properties of Discrete-Time Complex Exponentials10:40
Explore how discrete-time complex exponentials behave differently from continuous-time ones, showing that discrete-time signals e^{j ω0 n} are periodic only when ω0/2π is rational, unlike continuous time.
Example - 4 - Periodicity of a Discrete-Time Exponential10:24
Compute the fundamental period of a discrete-time signal composed of two exponentials by finding each component's period (3 and 8) and taking their least common multiple (24).
1.5.1 Discrete Time Unit Step and Unit Impulse Function12:18
Explore discrete time unit impulse and unit step functions as building blocks for signals, with definitions and graphs, and illustrate how impulse equals the first difference of the step.
1.5.2 Continuous Time Unit Step and Unit Impulse Function9:37
Explore continuous-time unit step and unit impulse functions, their relationship through differentiation and integration, and the delta function as an idealized limit.
Example - 5 - Unit Step and Unit Impulse Signals4:05
Explore how unit step and unit impulse functions model signal changes, derive the derivative, and identify discontinuities at t = 1, 2, and 4 with corresponding impulses.
1.6.1 Continuous and Discrete Time System Examples8:05
Explore continuous time and discrete time systems, mapping inputs to outputs in their time domains, with RC circuit and car dynamics illustrated by differential equations.
1.6.2 Interconnections of Systems14:05
Explore how interconnections of systems organize complex machines by linking subsystems in series, parallel, and feedback configurations, with block diagrams and real-world examples.
1.7.1 Basic System Properties - I22:30
Explains basic system properties, focusing on memory versus memoryless systems, with examples like accumulators and capacitors, and introduces causality and invertibility concepts.
1.7.2 Basic System Properties - II22:19
Explore stability, time invariance, and linearity in signals and systems, emphasizing bounded input and bounded output, and examine stable versus unstable examples and time-shift effects.
End of the Unit 1 Problems : 16:02
Explore end-of-unit one problems on converting complex numbers among rectangular, polar, and exponential forms, using magnitude and angle with examples like 1/2 e^{j pi} and sqrt(2) e^{-j pi/4}.
End of the Unit 1 Problems : 27:02
Explore expressing complex numbers in rectangular, polar, and exponential (phasor) forms within signals and systems unit one, including magnitude and phase calculations and division in phasor form.
End of the Unit 1 Problems : 38:37
Solve unit one problems by analyzing a piecewise signal x[n], determine its zero regions under shifts such as x[n-3], x[-n-2], and x[-n+2], and deduce the new zero intervals.
End of the Unit 1 Problems : 47:19
Apply the periodicity test y(t+T)=y(t) to two discrete-time examples, showing the first has zero period and is not periodic, while U[n] and U[-n] fail to repeat on the full axis.
End of the Unit 1 Problems : 511:57
Compute the even part of signals using x_even(n)=1/2[x(n)+x(-n)] for discrete and continuous cases, using unit step and sign functions to identify zero regions.

2.1 Intro5:24
Explore linear time invariant (LTI) systems and their representation with unit impulse signals, leveraging superposition and the convolution sum for discrete-time input-output analysis.
2.2.1 The Representation of Discrete-Time Signals in terms of Unit Impulses10:50
Learn how to represent discrete-time signals for linear time-invariant systems using shifted unit impulses, expressing any signal as a sum of delta functions weighted by X[k].
2.2.2 The Discrete Time Unit Impulse Response and Convolution-Sum Representation19:47
Represent any discrete-time input as a superposition of shifted unit impulses, and compute the output of a linear time-invariant system using convolution-sum with the unit impulse response.
2.2.2 Example - 16:54
this lecture demonstrates solving a discrete-time lti system using convolution, relating output y[n] to input x and impulse response h, and shifting h with x[0] and x[1] to compute y[n].
2.2.2 Example - 217:11
Learn how convolution uses a time-reversed and shifted impulse response with the input to produce the output. Analyze overlaps between x[k] and h[mx-k] to identify zero and nonzero regions.
2.2.2 Example - 310:03
Learn how discrete-time LTI systems use convolution to compute y[n] from x[k] and h[n], yielding y[n] = (1 - alpha^{n+1})/(1 - alpha) u[n].

3.1 Expressing the Idea in Mathematical Terms14:21
Learn how any periodic function can be expressed as an infinite sum of sine and cosine terms with coefficients a_n and b_n, plus a zero-term a_0.
3.2 Finding out the coefficients a0, an and bn25:02
Compute a0, an, and bn for a Fourier series on [0, T] using orthogonal functions. Use integrals to express coefficients with sine and cosine building blocks.
3.3 Some Observations on the Result (symmetry)12:49
Examine how Fourier series use sine and cosine terms, with symmetry determining a0, an, and bn, and limits shift by period T. Also note that a0 is the average value.
3.4 Example - 119:42
derive the Fourier series for f(x)=a x with period T, determine a0, an, bn via symmetry and integration by parts, and discuss convergence of sine and cosine terms.
3.4.1 Example - 214:30
Derive the Fourier series for a period-2 piecewise function with f(x)=x on [0,1] and 0 on [1,2]. Calculate a0, an, bn via integration by parts and assemble the Fourier series.
3.5 Complex Fourier Series22:05
Explore complex Fourier series by converting cosine and sine terms into exponential forms. Derive coefficients via Euler's relation and express the function as a sum from minus to plus infinity.
3.6 Complex Fourier series - 211:23
Compare complex and trig Fourier series; derive c_n from one integral and relate it to a_n, b_n, with c_0 = a_0 and c_{-n} = (a_n + i b_n)/2.
3.7 Review Fourier Series28:17
Review the Fourier series foundations using sine and cosine building blocks, derive amplitude, frequency, and phase from the trigonometric form, and relate them to the complex coefficients.
3.8 Example -325:47
This lecture derives the complex Fourier series for f(x), computes c_n, and uses Euler's formula to show it matches the trigonometric Fourier series for the given function.
4.1 Fourier Transform -124:05
Learn how Fourier transform builds on Fourier series to represent non-periodic functions as a continuous spectrum, using complex and trigonometric forms and coefficients c(k).
4.2 Fourier Transform - 216:08
Explore the Fourier transform and its comparison with Fourier series, including interpretation of C(K), amplitude, frequency, and phase, and how non-periodic signals relate to the continuous transform.

E.1 Fundamentals of Functions16:31
Explains periodic functions and their repeats every t, and introduces odd and even symmetries. Demonstrates sine and cosine as building blocks with amplitude, angular frequency, and phase concepts.
E.2 Fundamentals of Complex Numbers17:24
Explore complex numbers, the imaginary unit i, and rectangular and polar representations, then prove Euler's formula e^{i x} = cosine x + i sine x using Maclaurin series.
E.3 Orthogonal Functions24:21
Explore orthogonal functions and their dot-product intuition, showing that sine and cosine functions are perpendicular over a period, forming the backbone of Fourier series.
E.4 Orthogonal Functions and More Integrals13:21
Examine orthogonal functions via sine, cosine, and exponential integrals, showing when sine products vanish or yield L/2 and when exponentials yield zero or L for Fourier series.
E.5 The Single Period Idea13:47
Explore how sine and cosine serve as building blocks for signals, define fundamental period, and illustrate single-period and multiple-frequency ideas using harmonics like 2πt, 4πt, and 6πt.

Requirements

Basic Math

Description

In this course, we start our journey form discussing the general representation of signals and systems which is really important.

We will be representing our signals as mathematical functions having an independent variable. And this representation will be the most of the first unit.

Then, we are going to classify our signals, such as periodic/non-periodic, even/odd etc etc. This part will be also really crucial, since the kind of the signal allows us to choose the path to move on as we proceed to the solution.

Common Signals will be explained !!!

The Chapter for coming signals provides in-depth treatment of singularity functions such as unit pulse, unit step, and unit ramp signals. All of these signals are defined graphically and mathematically in the CT (Continuous Time) and DT (Discrete Time) domains. Signal properties and relationships between singularity functions are also explained. And other signals like signum function, sinc function etc. are also covered.

Properties of Systems: All system properties such as linearity, time invariance, causality, stability, memory and reversibility are well explained using a lot of examples.

You also have the opportunity to ask any question you have in your mind to the instructor 24/7.

We will be replying your messages and questions within 24 hours. (as soon as possible)

in Afterclap, We Trust

Sincerely,

kavcar

Afterclap Academy

Who this course is for:

For electrical engineering students taking "Signals and Systems"
For those, who would like to find out the bond-relationship between input and output.
For those, who would like to clarify system theory. Analysis of a signal.
Those, who are only interested and looking for sum fun.
Undergraduate engineering students with Electrical engineering, Electronics engineering, Biomedical engineering, Instrumentation engineering as specialisation
Diploma/Polytechnic students with Electronics engineering, Communication engineering, Instrumentation as specialisation

Signals and Sytems

What you'll learn

Explore related topics

Course content

1 - Signals and Systems Overview25 lectures • 4hr 26min

2 - Linear and Time Invariant Systems (LTI Systems)6 lectures • 1hr 10min

3 - Fourier series and Transform11 lectures • 3hr 34min

The Preparation you might need for Fourier series and transform5 lectures • 1hr 25min

Requirements

Description

Who this course is for: