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

Complex sinusoids are disproportionately emphasized among all signals in DSP formulations. Why? This is what we are going to investigate now.

## System input

Let’s probe a system that has an impulse response  $h[n]$ with a complex sinusoid $x[n]=e^{j\omega_0 n}$ as shown in the figure below:

- Keep in mind that the frequency response is the Fourier transform of the impulse response $h[n]$.
$$
H(\omega)=\sum_n h[n]e^{-j\omega n}
$$
- The system output is given by the convolution between the input $x[n]=e^{j\omega_0 n}$ and the impulse response $h[n]$.
$$
y[n]=\sum_m x[m]h[n-m]
$$

In such a simple setup, we plug in the value of $x[n]$ and see where it leads.
$$
\begin{align*}
y[n]&=\sum_m e^{j\omega_0 m}h[n-m]\\
&= \sum_m e^{j\omega_0 (m-n+n)}h[n-m]\\
&= e^{j\omega_0 n}\sum_m h[n-m]e^{-j\omega_0(n- m)}\\
&= e^{j\omega_0 n}H(\omega_0)
\end{align*}
$$
where the last step follows from a change of variable $q=n-m$.

So we have a surprising result here. The same complex sinusoid that was given as an input has come out as the output! But how is this complex sinusoid changed by the system? This is what we’ll investigate next.


## Impact of frequency response

In addition to the input as it is, the output has an additional component $H(\omega_0)$. Clearly, this is the system frequency response $H(\omega)$ *sampled at frequency $\omega_0$*.

Since $H(\omega_0)$ is a complex constant that has a magnitude and phase, the resultant expression can be written as:
$$
\begin{align}
y[n]&=e^{j\omega_0 n}|H(\omega_0)|e^{j\measuredangle H(\omega_0)}\\
&=|H(\omega_0)|\cdot e^{j\left\{\omega_0 n+\measuredangle H(\omega_0)\right\}}
\end{align}
$$

In other words, these two changes happen to that input sinusoid:
- Its magnitude is multiplied by the magnitude of system response at that frequency.
- The phase of system response at that frequency is added to its phase.

Now let's verify these findings with some code.

Complex sinusoids are disproportionately emphasized among all signals in DSP formulations. Why? This is what we are going to investigate now.

# System input

Let’s probe a system that has an impulse response  $h[n]$ with a complex sinusoid $x[n]=e^{j\omega_0 n}$ as shown in the figure below:

- Keep in mind that the frequency response is the Fourier transform of the impulse response $h[n]$.
$$
H(\omega)=\sum_n h[n]e^{-j\omega n}
$$
- The system output is given by the convolution between the input $x[n]=e^{j\omega_0 n}$ and the impulse response $h[n]$.
$$
y[n]=\sum_m x[m]h[n-m]
$$

In such a simple setup, we plug in the value of $x[n]$ and see where it leads.
$$
\begin{align*}
y[n]&=\sum_m e^{j\omega_0 m}h[n-m]\\
&= \sum_m e^{j\omega_0 (m-n+n)}h[n-m]\\
&= e^{j\omega_0 n}\sum_m h[n-m]e^{-j\omega_0(n- m)}\\
&= e^{j\omega_0 n}H(\omega_0)
\end{align*}
$$
where the last step follows from a change of variable $q=n-m$.

So we have a surprising result here. The same complex sinusoid that was given as an input has come out as the output! But how is this complex sinusoid changed by the system? This is what we’ll investigate next.


# Impact of frequency response

In addition to the input as it is, the output has an additional component $H(\omega_0)$. Clearly, this is the system frequency response $H(\omega)$ *sampled at frequency $\omega_0$*.

Since $H(\omega_0)$ is a complex constant that has a magnitude and phase, the resultant expression can be written as:
$$
\begin{align}
y[n]&=e^{j\omega_0 n}|H(\omega_0)|e^{j\measuredangle H(\omega_0)}\\
&=|H(\omega_0)|\cdot e^{j\left\{\omega_0 n+\measuredangle H(\omega_0)\right\}}
\end{align}
$$

In other words, these two changes happen to that input sinusoid:
- Its magnitude is multiplied by the magnitude of system response at that frequency.
- The phase of system response at that frequency is added to its phase.

Now let's verify these findings with some code.

Investigate the role of complex sinusoids to determine the frequency and impulse response of a system.

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 Role of Complex Sinusoids in System Characterization

System input