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

The DFT of a rectangular signal $x[n]$ is found to be a sinc signal.
$$
X[k]=\frac{\sin\left( \pi\frac{k}{N}L\right)}{\pi\frac{k}{N}}
$$

This sinc signal has a #key# main lobe:A lobe is the part of the spectrum that looks like an inverted parabola. #key# that is centered around the frequency bin $k = 0$. To find the amplitude at the y-axis that determines the height of the main lobe, we can't put $k=0$ in the expression above because both the numerator and denominator become zero, generating an indeterminate form.

## Main lobe peak

From the DFT definition, we have
$$
X[k]=\sum_{n=-\frac{N}{2}}^{\frac{N}{2}}x[n]e^{-j2\pi\frac{k}{N}n}=\sum_{n=-\frac{L-1}{2}}^{\frac{L-1}{2}}e^{-j2\pi\frac{k}{N}n}
$$
for an odd-length $L$ rectangular signal. Plug in $k=0$ at this stage to get:
$$
X[0]=\sum_{n=-\frac{L-1}{2}}^{\frac{L-1}{2}}1=L
$$
Interestingly, the value $X[0]$ is the sum of all the time domain samples and is known as its DC value if divided by $N$.

The DFT of a rectangular signal $x[n]$ is found to be a sinc signal.
$$
X[k]=\frac{\sin\left( \pi\frac{k}{N}L\right)}{\pi\frac{k}{N}}
$$

This sinc signal has a #key# main lobe:A lobe is the part of the spectrum that looks like an inverted parabola. #key# that is centered around the frequency bin $k = 0$. To find the amplitude at the y-axis that determines the height of the main lobe, we can't put $k=0$ in the expression above because both the numerator and denominator become zero, generating an indeterminate form.

# Main lobe peak

From the DFT definition, we have
$$
X[k]=\sum_{n=-\frac{N}{2}}^{\frac{N}{2}}x[n]e^{-j2\pi\frac{k}{N}n}=\sum_{n=-\frac{L-1}{2}}^{\frac{L-1}{2}}e^{-j2\pi\frac{k}{N}n}
$$
for an odd-length $L$ rectangular signal. Plug in $k=0$ at this stage to get:
$$
X[0]=\sum_{n=-\frac{L-1}{2}}^{\frac{L-1}{2}}1=L
$$
Interestingly, the value $X[0]$ is the sum of all the time domain samples and is known as its DC value if divided by $N$.

Derive the main properties of the spectrum in a rectangular signal.

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

Main Lobe Peak: Width and Zero Crossings

Main lobe peak