Hey it's a me again @drifter1!
Today we continue with my mathematics series about Signals and Systems in order to cover Exercises on Discrete-Time Fourier Series and Transform.
So, without further ado, let's dive straight into it!
Discrete-Time Fourier Series Coefficients Calculation [Based on 10.3 from Ref1]
Let's consider the following periodic sequence:
Find the appropriate expression for the envelope of the Fourier series coefficients and sample it.
In discrete-time an orthogonal pulse in the range [-N1, N1] can be represented as follows:
Using complex exponentials, the Fourier series coefficient is thus given as follows:
As such, the samples can be taken from the sampling function, which leads us to the following envelope:
In the case of this example, N1 = 2, and N = 10, and so:
Discrete-Time Fourier Transform Calculation [Based on 11.1 from Ref1]
Compute the discrete-time Fourier transform of the following signals:
The Fourier transform can be easily calculated with the help of Fourier transform pair tables and properties that we discussed throughout the series of articles.
The first case is a slighly different representation of one of the known Fourier transform pairs, which is:
For |a| > 1, this can be re-written as:
Thus, for a = 6, which is the first signal, the result is:
This case is quite similar to the first one.
Let's start re-formulating in order to achieve such a form:
From the previous example we know that:
Due to linearity its possible to multiply both sides by 36, which yields:
From the time-shifting property, we then get the final result:
The orthogonal pulse can be easily described using unit step functions, in the following manner:
The Fourier transform of the unit step function is:
And time-shifting by 3 leads to the following result:
So, the final result is:
Output of LTI System [Based on 11.2 from Ref1]
The following linear constant-coefficient difference equation describes an LTI system initially at rest:
Using Fourier transforms, let's evaluate y[n] for each of the following inputs:
First of all, let's take the Fourier transform of both sides of the LCCDE:
As such H(Ω) is equal to:
For x[n] = δ[n], X(Ω) = 1, which makes the output equal to the impulse response:
and, so the Inverse Fourier Transform gives us the output y[n] and h[n] in the same time:
The Fourier Transform of x2[n] is:
and so the output Y(Ω) is equal to:
Taking the Inverse Fourier Transform gives us the final result:
- Alan Oppenheim. RES.6-007 Signals and Systems. Spring 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA.
Mathematical equations used in this article were made using quicklatex.
Previous articles of the series
- Introduction → Signals, Systems
- Signal Basics → Signal Categorization, Basic Signal Types
- Signal Operations with Examples → Amplitude and Time Operations, Examples
- System Classification with Examples → System Classifications and Properties, Examples
- Sinusoidal and Complex Exponential Signals → Sinusoidal and Exponential Signals in Continuous and Discrete Time
LTI Systems and Convolution
- LTI System Response and Convolution → Linear System Interconnection (Cascade, Parallel, Feedback), Delayed Impulses, Convolution Sum and Integral
- LTI Convolution Properties → Commutative, Associative and Distributive Properties of LTI Convolution
- System Representation in Discrete-Time using Difference Equations → Linear Constant-Coefficient Difference Equations, Block Diagram Representation (Direct Form I and II)
- System Representation in Continuous-Time using Differential Equations → Linear Constant-Coefficient Differential Equations, Block Diagram Representation (Direct Form I and II)
- Exercises on LTI System Properties → Superposition, Impulse Response and System Classification Examples
- Exercise on Convolution → Discrete-Time Convolution Example with the help of visualizations
- Exercises on System Representation using Difference Equations → Simple Block Diagram to LCCDE Example, Direct Form I, II and LCCDE Example
- Exercises on System Representation using Differential Equations → Equation to Block Diagram Example, Direct Form I to Equation Example
Fourier Series and Transform
- Continuous-Time Periodic Signals & Fourier Series → Input Decomposition, Fourier Series, Analysis and Synthesis
- Continuous-Time Aperiodic Signals & Fourier Transform → Aperiodic Signals, Envelope Representation, Fourier and Inverse Fourier Transforms, Fourier Transform for Periodic Signals
- Continuous-Time Fourier Transform Properties → Linearity, Time-Shifting (Translation), Conjugate Symmetry, Time and Frequency Scaling, Duality, Differentiation and Integration, Parseval's Relation, Convolution and Multiplication Properties
- Discrete-Time Fourier Series & Transform → Getting into Discrete-Time, Fourier Series and Transform, Synthesis and Analysis Equations
- Discrete-Time Fourier Transform Properties → Differences with Continuous-Time, Periodicity, Linearity, Time and Frequency Shifting, Conjugate Summetry, Differencing and Accumulation, Time Reversal and Expansion, Differentation in Frequency, Convolution and Multiplication, Dualities
- Exercises on Continuous-Time Fourier Series → Fourier Series Coefficients Calculation from Signal Equation, Signal Graph
- Exercises on Continuous-Time Fourier Transform → Fourier Transform from Signal Graph and Equation, Output of LTI System
Final words | Next up
And this is actually it for today's post!
From next time we will start getting into concepts like Filtering, Modulation, Sampling, Interpolation etc.
Keep on drifting!