EECS20N: Signals and Systems

• Home
•  Up to Topics
•  Previous
•  Next

Week 11

• Imperative
• Impulses
• Sums of impulses
• Impulse response
• Convolution
• FIR
• Moving Average
• Convolution
• Causality
• Implementation
• Composition

Impulses

Intuitively, an impulse is a signal that is zero everywhere except at time zero. In the discrete-time case, δ : Integers → Reals

where for all n ∈ Integers,

.

This is called the Kronecker delta function. Its graph is shown below:

If you were able to run applets, you would have an impulse plot here.

We can construct a finite version of the Kronecker delta function by simply defining it over a finite range of time. Such a finite Kronecker delta function has the useful and interesting property that its Fourier series coefficients all have the same magnitude, as demonstrated by the following applet:

If you were able to run applets, you would have a KroneckerDelta demo here.

In this applet, the Kronecker delta function is delayed so that it is centered in the visible window. Assuming a finite length of 32 samples (4ms at an 8kHz sample rate), the magnitude of the Fourier series coefficients are shown above. Notice that they all have the same magnitude. This leads to the interpretation that the delta function contains all frequencies in equal amounts.

The continuous-time case, which is called the Dirac delta function, is mathematically much more difficult to work with. Like the Kronecker delta function, it is zero everywhere except at zero. But unlike the Kronecker delta function, its value is infinite at zero. We will not study its subtleties, but rather just introduce it and assert some results without fully demonstrating their validity. The Dirac delta function is defined to be

δ :Reals → Reals₊₊

where Reals₊₊ = Reals ∪ {∞ }, and for all t ∈ Reals where t ≠ 0,

δ (t) = 0,

and where the following property is satisfied,

.

For the latter property to be sastified, clearly no finite value at t = 0 would suffice. This is why the value must be infinite at t = 0. Notice that the Kronecker delta function has a similar property,

,

but that in this case, the property is trivial. There is no mathematical subtlety.

The following applet shows finite Fourier series approximations to a Dirac delta function. The Dirac delta function, of course, cannot be represented precisely, because its infinitely narrow width and infinite height are problematic for the computer. The following applet approximates the ideal by showing a fairly narrow and tall triangular pulse.

If you were able to run applets, you would have a DiracDelta demo here.

Notice that as more Fourier series components are added to the approximation, the approximation improves. Moreover, each component has equal magnitude, which again leads to the interpretation that a Dirac delta function contains all frequency components in equal amounts. Unlike the discrete-time case, however, it includes all multiples of the fundamental frequency out to infinity. This is the source of the mathemtical subtleties in dealing with it.