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The Impulse/Dirac Distribution

Module by: Richard Baraniuk. E-mail the author

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

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

Generalize

To a broader family of distributions δ t o ⁢s⁢t≡s⁢ t o δ t o s t s t o Alternate notation: ∫−∞∞δ⁢t− t o ⁢s⁢tdt =s⁢ t o t δ t t o s t s t o

Intuition

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

Example 1

δ⁢t=limit   ε ε→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)⇒δ⁢t 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:x⁢n=1 n x n 1

Immediate realization:

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

Figure 3
Divine Intervention
Divine Intervention (fig3.png)

Impulse Distribution

X⁢ei⁢ω=∑ r =−∞∞2⁢π⁢δ⁢ω+2⁢π⁢r X ω r 2 δ ω 2 r (3)
Figure 4
Figure 4 (fig4.png)

1.a) Verify

Formally use DTFT formula x⁢n⁢≟⁢∫−ππX⁢ei⁢ω⁢ei⁢ω⁢n⁢12⁢πdω =∫−ππ(2⁢π)⁢∑r =−∞∞δ⁢ω+2⁢π⁢r⁢ei⁢ω⁢n⁢12⁢πdω 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.
=2⁢π2⁢π⁢ei⁢ω⁢n| ω =00=1=x⁢n ∀n:n ω 0 0 2 2 ω n 1 x n n This "comb" of impulses works in the formula.

1.b) Intuition

  • x⁢n 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 x⁢n=ei⁢ ω o ⁢n ∀n:n x n ω o n n "pure frequency"

then X⁢ei⁢ω=∑ r =−∞∞δ⁢ω− ω o +2⁢π⁢r 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 ⁢n⁢e(−i)⁢ω⁢n=∑ r =−∞∞(2⁢π)⁢δ⁢ω− ω o +2⁢π⁢r n ω o n ω n r 2 δ ω ω o 2 r

Interesting:

( ω o =0)→(∑ n =−∞∞e(−i)⁢ω⁢n=∑ r =−∞∞(2⁢π)⁢δ⁢ω+2⁢π⁢r) ω o 0 n ω n r 2 δ ω 2 r "sum of a sinusiod = an impulse combination" (Reference)

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