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

To understand the idea of orthogonality, we consider both real and complex signals.

## Real sinusoids

The figure below shows the continuous-time versions of two sinusoids containing $N$ samples, one with frequency $f_s\cdot 1/N$ and the other with frequency $f_s\cdot 2/N$. The discrete frequencies are $1/N$ and $2/N$, but it's easier to understand this idea with a continuous curve than with a discrete sinusoid.

Observe the "Q" part (the vertical plane at the back of the screen) of the two waves and look at the number of samples in each.
- The wave with frequency $1/N$ has one cycle in $N$ samples.

- The wave with frequency $2/N$ has one cycle in $N$ samples.

Clearly, the wave with frequency $2/N$ completes one cycle for the positive half of the first sinusoid and another cycle for the negative half of the first sinusoid. Due to these opposite signs, a summation of their sample-by-sample product is zero.
**Orthogonality** between two signals means that a summation of their sample-by-sample product (i.e., correlation at time $0$) is zero.

Mathematically, we write for the "Q" part in the figure above:
$$    
\begin{equation*} 
            \sum \limits _{n=0} ^{N-1} \sin 2\pi \frac{1}{N}n \cdot \sin 2\pi \frac{2}{N}n = 0
    \end{equation*}
$$

Let's explore this same idea with complex signals.

## Complex sinusoids

For a complex sinusoid, we have:
$$
\begin{equation*} 
            \sum \limits _{n=0} ^{N-1} e^{j 2\pi \frac{1}{N}n} \cdot e^{-j 2\pi \frac{2}{N}n} = 0
    \end{equation*}
$$
Being a complex conjugate, the second term above has a negative power because the expression for correlation of two complex signals involves the conjugation of the second signal.
- The product of the above two signals is $e^{-j2\pi\frac{1}{N}n}$: a complex sinusoid with a negative frequency of $1/N$.
- Applying the summation generates a zero output because both cosine (the real part) and sine (the imaginary part) have one full cycle in those $N$ samples. The negative halves cancel out the positive halves.

Extending this idea further, two complex sinusoids with frequencies $k_1/N$ and $k_2/N$ are orthogonal when
$$
\begin{equation*}
\begin{aligned}
            \sum _{n=0}^{N-1}e^{j 2\pi \frac{k_1-k_2}{N}n} &= \begin{cases}
                                                            N &  k_1 = k_2 \\
                                                            0 & \text{otherwise}
                                                        \end{cases}          
\end{aligned}
\end{equation*}
$$

where $k_1$ and $k_2$ are integers from $0$ to $N-1$.

In words, a correlation of a sinusoid with itself is maximum and zero with other sinusoids.
This concept is illustrated in the code below:

# Real sinusoids

The figure below shows the continuous-time versions of two sinusoids containing $N$ samples, one with frequency $f_s\cdot 1/N$ and the other with frequency $f_s\cdot 2/N$. The discrete frequencies are $1/N$ and $2/N$, but it's easier to understand this idea with a continuous curve than with a discrete sinusoid.

Observe the "Q" part (the vertical plane at the back of the screen) of the two waves and look at the number of samples in each.
- The wave with frequency $1/N$ has one cycle in $N$ samples.

- The wave with frequency $2/N$ has one cycle in $N$ samples.

Clearly, the wave with frequency $2/N$ completes one cycle for the positive half of the first sinusoid and another cycle for the negative half of the first sinusoid. Due to these opposite signs, a summation of their sample-by-sample product is zero.
**Orthogonality** between two signals means that a summation of their sample-by-sample product (i.e., correlation at time $0$) is zero.

Mathematically, we write for the "Q" part in the figure above:
$$    
\begin{equation*} 
            \sum \limits _{n=0} ^{N-1} \sin 2\pi \frac{1}{N}n \cdot \sin 2\pi \frac{2}{N}n = 0
    \end{equation*}
$$

Let's explore this same idea with complex signals.

# Complex sinusoids

For a complex sinusoid, we have:
$$
\begin{equation*} 
            \sum \limits _{n=0} ^{N-1} e^{j 2\pi \frac{1}{N}n} \cdot e^{-j 2\pi \frac{2}{N}n} = 0
    \end{equation*}
$$
Being a complex conjugate, the second term above has a negative power because the expression for correlation of two complex signals involves the conjugation of the second signal.
- The product of the above two signals is $e^{-j2\pi\frac{1}{N}n}$: a complex sinusoid with a negative frequency of $1/N$.
- Applying the summation generates a zero output because both cosine (the real part) and sine (the imaginary part) have one full cycle in those $N$ samples. The negative halves cancel out the positive halves.

Extending this idea further, two complex sinusoids with frequencies $k_1/N$ and $k_2/N$ are orthogonal when
$$
\begin{equation*}
\begin{aligned}
            \sum _{n=0}^{N-1}e^{j 2\pi \frac{k_1-k_2}{N}n} &= \begin{cases}
                                                            N &  k_1 = k_2 \\
                                                            0 & \text{otherwise}
                                                        \end{cases}          
\end{aligned}
\end{equation*}
$$

where $k_1$ and $k_2$ are integers from $0$ to $N-1$.

In words, a correlation of a sinusoid with itself is maximum and zero with other sinusoids.
This concept is illustrated in the code below:

Explore why many DSP algorithms rely on orthogonality.

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

The Idea of Orthogonality

Real sinusoids