Discover digital signal processing fundamentals, delve into signal design and analysis, learn about transforms, and develop practical OFDM systems focusing on modulation and demodulation. Gain insights into time and frequency domains.

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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

An ADC samples a continuous-time signal to produce discrete-time samples. From the viewpoint of a digital signal processor, this signal is represented in computer memory as a series of numbers. Unless the sample rate $f_s$ is known, there's little that can be done.

## Determining the frequency

With the available information about $f_s$, we can discover the frequency by first computing the period.
For example, the figure below shows a discrete-time sinusoid with a period of $20$ samples. In terms of real time in seconds, this period depends on $f_s$.

- $f_s=10$ Hz implies that $T_s = 0.1$ seconds, so:
$$
    \begin{align*}
        T &= 20 ~\frac{\textmd{samples}}{\textmd{period}}~ \cdot ~ 0.1 ~\frac{\textmd{seconds}}{\textmd{sample}} = 2~ \textmd{seconds}
    \end{align*}
$$    
Therefore, the frequency $f$ is given by:
$$
f = \frac{1}{2}=0.5 ~\text{Hz}
$$
- $f_s=100$ Hz implies that $T_s = 0.01$ seconds, so:
$$
    \begin{align*}
        T &= 20 ~\frac{\textmd{samples}}{\textmd{period}}~ \cdot ~0.01 ~\frac{\textmd{seconds}}{\textmd{sample}} = 0.2~ \textmd{seconds}
    \end{align*}
$$
Therefore, the frequency $f$ is given by:
$$
f = \frac{1}{0.2}=5 ~\text{Hz}
$$

The main point is that there is no difference between the two signals above while they reside in the processor memory. Let’s explore this idea further.

## Discrete frequency

Just like a continuous-time signal cannot be stored in a computer, a continuous-frequency spectrum is impossible to display for any digital machine. Therefore, a sample from the frequency domain is also desired.

Assume that there are a total of $N$ samples available for this purpose. At a rate of $f_s$, the lowest frequency comes from the longest period. And the longest period is $NT_s$ seconds (1 full cycle). Therefore, the smallest frequency is:
$$
f = \frac{1}{NT_s}=\frac{f_s}{N}
$$

The actual complex sinusoid is given by:
$$
e^{j2\pi \frac{f_s}{N}t}\bigg|_{t=nT_S}=e^{j2\pi \frac{f_s}{N}nT_s}=e^{j2\pi \frac{1}{N}n}
$$

In the expression above, $1/N$ is the discrete frequency of this sinusoid. Integer multiples of $1/N$ can be used for a total of $N$ discrete frequencies. We can understand this further by looking at the code given below:





An ADC samples a continuous-time signal to produce discrete-time samples. From the viewpoint of a digital signal processor, this signal is represented in computer memory as a series of numbers. Unless the sample rate $f_s$ is known, there's little that can be done.

# Determining the frequency

With the available information about $f_s$, we can discover the frequency by first computing the period.
For example, the figure below shows a discrete-time sinusoid with a period of $20$ samples. In terms of real time in seconds, this period depends on $f_s$.

- $f_s=10$ Hz implies that $T_s = 0.1$ seconds, so:
$$
    \begin{align*}
        T &= 20 ~\frac{\textmd{samples}}{\textmd{period}}~ \cdot ~ 0.1 ~\frac{\textmd{seconds}}{\textmd{sample}} = 2~ \textmd{seconds}
    \end{align*}
$$    
Therefore, the frequency $f$ is given by:
$$
f = \frac{1}{2}=0.5 ~\text{Hz}
$$
- $f_s=100$ Hz implies that $T_s = 0.01$ seconds, so:
$$
    \begin{align*}
        T &= 20 ~\frac{\textmd{samples}}{\textmd{period}}~ \cdot ~0.01 ~\frac{\textmd{seconds}}{\textmd{sample}} = 0.2~ \textmd{seconds}
    \end{align*}
$$
Therefore, the frequency $f$ is given by:
$$
f = \frac{1}{0.2}=5 ~\text{Hz}
$$

The main point is that there is no difference between the two signals above while they reside in the processor memory. Let’s explore this idea further.

# Discrete frequency

Just like a continuous-time signal cannot be stored in a computer, a continuous-frequency spectrum is impossible to display for any digital machine. Therefore, a sample from the frequency domain is also desired.

Assume that there are a total of $N$ samples available for this purpose. At a rate of $f_s$, the lowest frequency comes from the longest period. And the longest period is $NT_s$ seconds (1 full cycle). Therefore, the smallest frequency is:
$$
f = \frac{1}{NT_s}=\frac{f_s}{N}
$$

The actual complex sinusoid is given by:
$$
e^{j2\pi \frac{f_s}{N}t}\bigg|_{t=nT_S}=e^{j2\pi \frac{f_s}{N}nT_s}=e^{j2\pi \frac{1}{N}n}
$$

In the expression above, $1/N$ is the discrete frequency of this sinusoid. Integer multiples of $1/N$ can be used for a total of $N$ discrete frequencies. We can understand this further by looking at the code given below:





Discover how sampling a signal leads to a new set of frequencies.

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

Sample Rate and Discrete Frequency

Determining the frequency