This course focuses on the fundamentals of digital signal processing that covers the fundamentals of signals, systems and transforms.

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We live in a world filled with signals which can be represented by sequences of numbers in a digital machine. These sequences of numbers can be manipulated by algorithms and have numerous applications in audio, images, video, wireless transmission and countless other domains. 

In this course, you’ll understand the field of digital signal processing with a focus on the design and analysis of signals, and algorithms. You’ll cover the DSP fundamentals of signals, systems and transforms. You’lll learn about the basic signal types and perform different operations. You’ll develop an understanding about the frequency domain and how to represent signals in time, and the transformation from one domain to the other using the Fourier transform. You’ll also learn about digital systems. Finally, you’ll develop a practical OFDM system focusing on modulation and demodulation.

By the end of this course, you’ll be able to model, analyze and design signals and systems in greater depth, in the time and frequency domains.

Fundamentals of Digital Signal Processing

It's a linear phase plot as a function of frequency.

The DFT of a rectangular signal symmetric around time $0$ is a sinc signal with a zero phase. What happens to the phase plot for a rectangular signal shifted in time (e.g., from $n=0$ to $n=L$)?

The DFT of a rectangular signal symmetric around time <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>0 is a sinc signal with a zero phase. What happens to the phase plot for a rectangular signal shifted in time (e.g., from <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n=0</annotation></semantics></math>n=0 to <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">n=L</annotation></semantics></math>n=L)?

Consider the following discrete-time signal:
$$
x[n]= \{3,-1,2,-7\}
$$
Without computing the DFT $X[k]$ explicitly and for $N=4$, find the following:
- $X[0]$
- $X[2]$

Consider the following discrete-time signal:
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">{</mo><mn>3</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>−</mo><mn>7</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">x[n]= \{3,-1,2,-7\}
</annotation></semantics></math>x[n]={3,−1,2,−7}
Without computing the DFT <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">X[k]</annotation></semantics></math>X[k] explicitly and for <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">N=4</annotation></semantics></math>N=4, find the following:
<ul>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">X[0]</annotation></semantics></math>X[0]</li>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">X[2]</annotation></semantics></math>X[2]</li>
</ul>

Determine the 30-point DFT of the following sequence without computing the DFT exactly:
$$
x[n]=15e^{j2\pi n/3}, \quad n =0,1,\cdots,29
$$

Determine the 30-point DFT of the following sequence without computing the DFT exactly:
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mn>15</mn><msup><mi>e</mi><mrow><mi>j</mi><mn>2</mn><mi>π</mi><mi>n</mi><mi mathvariant="normal">/</mi><mn>3</mn></mrow></msup><mo separator="true">,</mo><mspace width="1em"/><mi>n</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mn>29</mn></mrow><annotation encoding="application/x-tex">x[n]=15e^{j2\pi n/3}, \quad n =0,1,\cdots,29
</annotation></semantics></math>x[n]=15ej2πn/3,n=0,1,⋯,29

A sinc signal is given by:
$$
x(t) = \frac{\sin 2\pi f_0t}{2\pi f_0t}
$$
Find the location of the first zero crossing.

A sinc signal is given by:
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>sin</mi><mo>⁡</mo><mn>2</mn><mi>π</mi><msub><mi>f</mi><mn>0</mn></msub><mi>t</mi></mrow><mrow><mn>2</mn><mi>π</mi><msub><mi>f</mi><mn>0</mn></msub><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">x(t) = \frac{\sin 2\pi f_0t}{2\pi f_0t}
</annotation></semantics></math>x(t)=2πf0​tsin2πf0​t​
Find the location of the first zero crossing.

Normalization isn't necessary in digital system implementations.

The DFT amplifies the contribution of each complex sinusoid by $N$, and the IDFT brings the amplitude back.

Why is there a normalization factor of $1/N$ in an IDFT?

Why is there a normalization factor of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mi>N</mi></mrow><annotation encoding="application/x-tex">1/N</annotation></semantics></math>1/N in an IDFT?

Introduction to Signals

Basic Signals and Operations

Complex Signals and the Frequency Domain

Sampling a Signal

The Discrete Fourier Transform

Fourier Transform Properties

Digital Systems

Putting It All Together: An OFDM Modem

Demodulating an OFDM Signal Stream

The Last Words

Bibliography

Quiz 3