DELTA

Returns 1 if two numbers are equal, and 0 otherwise.

Syntax:

DELTA(number1, number2)

number1 and number2 are numbers. If number2 is omitted it is assumed to be 0. This function is an implementation of the (mathematical) Kronecker delta function. number1=number2 returns TRUE or FALSE instead of 1 or 0, but is otherwise identical for number arguments.

Example:

DELTA(4, 5)

returns 0.

DELTA(4, A1)

where cell A1 contains 4, returns 1.

DELTA(0)

returns 1.

=4=5

in a cell shows FALSE, as 4 is not equal to 5.

Application:

A Digital Audio Equalizer

Imagine you are designing a simple digital audio equalizer. You have a stream of digital audio data, which is just a sequence of numbers (samples) representing the sound amplitude at discrete moments in time. The equalizer needs to modify the amplitude of a specific frequency component, but you want to test how the system responds to a single, isolated "click" or "pop" in the audio. This isolated event can be modeled by the Kronecker delta function.

The Problem

We want to send a single, unit-amplitude pulse into the system at a specific time step, let's say at n=5, and see how the equalizer's filters and amplifiers react. This is often called a "unit impulse response" test in signal processing.

The Mathematical Model

The input signal, x[n], can be modeled by the Kronecker delta function. We can write the signal as:

x[n]=δn5

Where the variable n represents the discrete time step (an integer). The Kronecker delta ensures that the signal has a value of 1 only at n=5 and is zero at all other time steps.

Table: Input Signal Values Over Time

	Discrete Time Step, n	Value of Kronecker Delta, x[n] = δn5	Description
	A	B	C
1	n = 1	0	No signal
2	n = 2	0	No signal
3	n = 3	0	No signal
4	n = 4	0	No signal
5	n = 5	1	The impulse or "click" occurs
6	n = 6	0	No signal
7	n = 7	0	No signal
8	...	0	...

Conclusion

By modeling the input with a Kronecker delta function, we are providing the simplest possible stimulus to the system. The output of the equalizer, which we could call y[n], would then be the "impulse response" of the system. Analyzing this response helps engineers understand how the equalizer will affect any other, more complex audio signal. The Kronecker delta is a powerful tool for simplifying these discrete-time problems, much like the Dirac delta is for continuous-time systems.