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The Impulse/Dirac Distribution

Module by: Richard Baraniuk. E-mail the author

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Summary: An overview of the concepts behind the Impulse/Dirac Distribution.

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Definition 1: Distribution
maps a function to a number
Definition 2: Impulse Distribution
is given by
δst≡s0 δ s t s 0 (1)
Often we write δ δ as δt δ t and use notation
∫-∞∞δtstdt=s0 t δ t s t s 0 (2)
Special case: ∫-∞∞δtdt=1 t δ t 1 . Why?

Generalize

To a broader family of distributions δ t o st≡s t o δ t o s t s t o Alternate notation: ∫-∞∞δt− t o stdt=s t o t δ t t o s t s t o

Intuition

We can think of δt δ t as the limit of tall narrow pulses of area 1.

Example 1

δt=limε→0 p ε t δ t ε ε 0 p ε t

Figure 1
Figure 1 (fig1.png)

But of course this limit does not exist at t=o⇒δt t o δ t is not a function.

Figure 2
Figure 2 (fig2.png)

Key Application if the Impulse Distribution to the DTFT

Exercise 1

Find the DTFT of ∀n:xn=1 n x n 1

Immediate realization:

since xn x n is not absolutely summable, we might expect something weird for an answer.

Figure 3
Divine Intervention
Divine Intervention (fig3.png)

Impulse Distribution

Xⅇⅈω=∑r=-∞∞2πδω+2πr X ω r 2 δ ω 2 r (3)
Figure 4
Figure 4 (fig4.png)

1.a) Verify

Formally use DTFT formula xn≟∫-ππXⅇⅈωⅇⅈωn12πdω=∫-ππ2π∑r=-∞∞δω+2πrⅇⅈωn12πdω x n ≟ ω X ω ω n 1 2 ω 2 r δ ω 2 r ω n 1 2

Note:
The integral picks up the impulse at ω=0 ω 0 and evaluates.
=2π2πⅇⅈωn|ω=00= 1 =xn ∀n ω 0 0 2 2 ω n 1 x n n This "comb" of impulses works in the formula.

1.b) Intuition

  • xn x n is pure "DC" therefore it is the purest possible low frequency signal.
  • But why an infinite sum of δ δ distributions?

1.c) Generalization

If xn=ⅇⅈ ω o n ∀n x n ω o n n "pure frequency"

then Xⅇⅈω=∑r=-∞∞δω− ω o +2πr X ω r δ ω ω o 2 r

Figure 5
Figure 5 (fig5.png)

Note:

Distribution theory thus gives a special meaning to the statement ∑n=-∞∞ⅇⅈ ω o nⅇ-ⅈωn=∑r=-∞∞2πδω− ω o +2πr n ω o n ω n r 2 δ ω ω o 2 r

Interesting:

ω o =0→∑n=-∞∞ⅇ-ⅈωn=∑r=-∞∞2πδω+2πr ω o 0 n ω n r 2 δ ω 2 r "sum of a sinusiod = an impulse combination" (Reference)

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