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

A company produces an ADC that can sample a signal at a rate of $f_s=1$ GHz. What is the time duration between every two samples?


A company produces an ADC that can sample a signal at a rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>s</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">f_s=1</annotation></semantics></math>fs​=1 GHz. What is the time duration between every two samples?

A company produces an ADC that can sample a signal at a rate of $f_s=1$ GHz. For a memory size of 2 MB and a word length of 1 byte for each sample, what is the maximum duration over which a signal can be stored?

A company produces an ADC that can sample a signal at a rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>s</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">f_s=1</annotation></semantics></math>fs​=1 GHz. For a memory size of 2 MB and a word length of 1 byte for each sample, what is the maximum duration over which a signal can be stored?

An analog signal is given by:
$$
x(t) = 3 \cos (2000\pi t) + 5\sin (6000 \pi t)+10\cos (12000\pi t)
$$
What is the minimum sample rate $f_s$ that satisfies the sampling theorem?


An analog 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><mn>3</mn><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>2000</mn><mi>π</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mn>5</mn><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>6000</mn><mi>π</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mn>10</mn><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>12000</mn><mi>π</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x(t) = 3 \cos (2000\pi t) + 5\sin (6000 \pi t)+10\cos (12000\pi t)
</annotation></semantics></math>x(t)=3cos(2000πt)+5sin(6000πt)+10cos(12000πt)
What is the minimum sample rate <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">f_s</annotation></semantics></math>fs​ that satisfies the sampling theorem?

An analog signal is given by:
$$
x(t) = 3 \cos (2000\pi t) + 5\sin (6000 \pi t)+10\cos (12000\pi t)
$$

Instead, if it is sampled at $f_s=5000$ Hz, what will be the three actual frequencies in the aliased signal?

An analog 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><mn>3</mn><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>2000</mn><mi>π</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mn>5</mn><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>6000</mn><mi>π</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mn>10</mn><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>12000</mn><mi>π</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x(t) = 3 \cos (2000\pi t) + 5\sin (6000 \pi t)+10\cos (12000\pi t)
</annotation></semantics></math>x(t)=3cos(2000πt)+5sin(6000πt)+10cos(12000πt)
Instead, if it is sampled at <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>s</mi></msub><mo>=</mo><mn>5000</mn></mrow><annotation encoding="application/x-tex">f_s=5000</annotation></semantics></math>fs​=5000 Hz, what will be the three actual frequencies in the aliased signal?

A digital signal stored in a processor memory completes 1 cycle in 50 samples. If the original sample rate was $f_s=1$ kHz, what is the time period in seconds?

A digital signal stored in a processor memory completes 1 cycle in 50 samples. If the original sample rate was <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>s</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">f_s=1</annotation></semantics></math>fs​=1 kHz, what is the time period in seconds?

If the output of an ADC at a sample rate of $f_s=8$ kHz is the signal
$$
x[n]= 4 \cos (0.5\pi n -\pi/5),
$$
find one possible input frequency of $x(t)$.

If the output of an ADC at a sample rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>s</mi></msub><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">f_s=8</annotation></semantics></math>fs​=8 kHz is the 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><mn>4</mn><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>0.5</mn><mi>π</mi><mi>n</mi><mo>−</mo><mi>π</mi><mi mathvariant="normal">/</mi><mn>5</mn><mo stretchy="false">)</mo><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">x[n]= 4 \cos (0.5\pi n -\pi/5),
</annotation></semantics></math>x[n]=4cos(0.5πn−π/5),
find one possible input frequency of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x(t)</annotation></semantics></math>x(t).

If a signal $x(t)$ is given as
$$
x(t)= 4 + 3\cos (1500\pi t-\pi/3)-8\cos(2800\pi t+\pi/4)
$$
what should be the sampling rate of $f_s$ be so that no distortion occurs?

If a signal <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x(t)</annotation></semantics></math>x(t) is given as
<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><mn>4</mn><mo>+</mo><mn>3</mn><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>1500</mn><mi>π</mi><mi>t</mi><mo>−</mo><mi>π</mi><mi mathvariant="normal">/</mi><mn>3</mn><mo stretchy="false">)</mo><mo>−</mo><mn>8</mn><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>2800</mn><mi>π</mi><mi>t</mi><mo>+</mo><mi>π</mi><mi mathvariant="normal">/</mi><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x(t)= 4 + 3\cos (1500\pi t-\pi/3)-8\cos(2800\pi t+\pi/4)
</annotation></semantics></math>x(t)=4+3cos(1500πt−π/3)−8cos(2800πt+π/4)
what should be the sampling rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">f_s</annotation></semantics></math>fs​ be so that no distortion occurs?

Test your knowledge about sampling and its impact on the signal spectrum.

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 2