# Tim's Blackboard #2: The Dirac Delta Misnomer

Blog Post created by Tim Wang Lee on Jun 7, 2017

Welcome to Tim’s Blackboard! This is the place to find discussions on interesting topics related to signal integrity and power integrity.

This week on Tim’s Blackboard is the “Dirac Delta Misnomer.”

# Introduction

Did you know the famous Dirac delta function is mathematically NOT a function? Fig. 1: ADS data display representation of the Dirac delta “function” by a line and arrowhead. The tip of the arrow head indicates the multiplicative constant to the Dirac delta.

The "function", shown in Fig. 1, can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite: and it is also constrained to satisfy the following identity : However, there is no function that can simultaneously have all the above properties.
Any extended-real function that is equal to zero everywhere but a single point must have total integral zero  .

Well, if Diract delta is not a function, what is it?

# Dirac Delta Distribution

Mathematically speaking, the Dirac delta “function” is a generalized function or a distribution. It can be considered as the limit of a zero-mean normal distribution when the standard deviation, σ, approaches 0, see Fig. 2. Fig. 2: Normal distribution with different standard deviations shown in ADS data display. As the value of standard deviation gets smaller and smaller, the function approaches Dirac delta distribution.

To rigorously capture the notion of the Dirac delta “function”, Mathematicians had also defined a measure. Instead of spending time on the mathematics, we will look at what brings the misnamed Dirac delta distribution to its fame.

# Dirac Delta and Impulse Response

The Dirac delta distribution is well known for many reasons. For example,

1.    The convolution of a Dirac delta function with function F is the original function F, ,

2.    The Fourier Transform of a Dirac delta is unity, , and most importantly,

3.    The response of a given Linear Time-Invariant (LTI) system to a Dirac delta distribution completely characterizes the system.

To build up a good foundation for future discussions on Convolution and Fourier Transform, let’s examine the impulse response: the response of an LTI system to the Dirac delta distribution. (note: in the following sections, the term impulse is used interchangeably with Dirac delta distribution.)

# Impulse Response of a Lossless Channel

Before we bring out simulation tools to simulate the impulse response of a lossless channel, it is important to know what to expect. Dr. Eric Bogatin named the practice Rule #9: “Never perform a measurement or simulation without first anticipating the results you expect to see.”

## Expectation

Shown in Fig. 3 is an illustration of the circuit setup. Given a lossless line with time delay, TD, and no mismatch to create reflection, we will see the impulse at the probe TD seconds after the impulse (Dirac delta distribution) is sent. Fig. 3: Illustration of sending an impulse through a transmission line with time delay TD seconds

In this experiment, we used a section of lossless transmission line that is 1 nsec long. Per Rule #9, we should expect the impulse to arrive at the output at 1 nsec.

As shown in Fig. 4, the simulation result is consistent with our expectation. The same impulse indeed shows up at the output 1 nsec after it leaves the source. Fig. 4: ADS simulation of an impulse through a 1 nsec long lossless transmission line. As expected, the impulse is delayed by 1 nsec.

Note that because it is impossible to generate an ideal Dirac delta distribution having an infinite amplitude, we are using the arrowhead to denote infinity.In the ADS workspace attached, you will find a method to approximate the impulse. The key is to ensure the approximated Dirac delta distribution has an integral of unity over the entire real line.

## Consistency Test

To make sure our approximated impulse satisfies the constraints formulated before, we also plot the integral of the approximated impulse. We would expect the plot of the integral of the output impulse to be a unit step function starting at 1 nsec. Fig. 5: The integral of the approximated impulse is indeed the step function we expected.

As shown in Fig. 5, the integral of the approximated impulse fulfills the unit step requirement. We now have confidence in the approximated impulse and the generated impulse response.

# Impulse Response of a Lossy Channel

In real life, a lossless channel does not exist. There is always conductor loss and/or dielectric loss in the transmission line. We will now investigate a 6 inch 50 Ohm microstrip line on an FR4 substrate with a virtual prototype.

## Expectation

Using the rule of thumb, 6 inch/nsec, for the speed of propagation in FR4, we would expect 1 nsec delay for a 6 inch transmission line. In addition, because of the frequency-dependent loss, we would also expect the impulse to spread out. Lastly, we should see a very high voltage peak approximating the infinite amplitude.

## ADS Simulation Setup and Result

In the lossy case simulation, we used the multilayer layer library substrate and transmission line so a 2D cross-section of the trace is solved by method of moments to gain more accuracy in simulating losses than the equation-based model.

In Fig. 6, the result of the simulation agrees with most of our predictions. The peak of the output voltage is more than 60 GV (daunting) and the impulse is more spread out because of the frequency-dependent loss. Fig. 6: The result of lossy transmission line simulation agrees with our predictions, except for the arrival time.

Nonetheless, although close to our prediction, the arrival time of the impulse is a bit off. Instead of 1 nsec, the impulse arrives at 0.88 nsec. (Any guesses on why that is? Feel free to post possible explanations in the comment section, and check back next week for the answer.)

# Dirac Delta Misnomer Corrected

After our journey today, we now know that because of its unique definition, the Dirac delta should to refer to as distribution and not a function, a fun fact to bring up at social functions (pun intended).

Moreover, we touched upon the important properties of the Dirac delta distribution. Specifically, the response of an LTI system to a Dirac delta: the impulse response, which characterizes an LTI system completely.

In future posts, we will build upon the impulse response idea and delve into Convolution and Fourier Transform.

That's this week's Tim's Blackboard. See you in two weeks!

Before then, make sure to download the workspace attached to see how an impulse can be approximated and how the impulse looks like after going through a realistic transmission line! # References

 Gel'fand, I. M.; Shilov, G. E. (1966–1968), Generalized functions, 1–5, Academic Press

 Vladimirov, V. S. (1971), Equations of mathematical physics, Marcel Dekker, ISBN 0-8247-1713-9.

 Duistermaat, Hans; Kolk (2010), Distributions: Theory and applications, Springer.