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More FS Properties

Module by: Richard Baraniuk

Summary: An overview a several more FS properties.

f ↔ ℱ.S. c n f ↔ ℱ.S. c n g↔ d n g ↔ d n

Signal Multiplication

ftgt=yt ↔ ℱ.S. ∑k=-∞∞ c k d n - k = e n f t g t y t ↔ ℱ.S. k c k d n - k e n

Note:

Discrete time convolution

Proof

e n =1T∫0Tftgtⅇ-ⅈ ω o ntdt=1T∫0T∑k=-∞∞ c k ⅇⅈ ω o ktgtⅇ-ⅈ ω o ntdt=∑k=-∞∞ c k 1T∫0Tgtⅇ-ⅈ ω o n-ktdt=∑k=-∞∞ c k d n - k e n 1 T t 0 T f t g t ω o n t 1 T t 0 T k c k ω o k t g t ω o n t k c k 1 T t 0 T g t ω o n k t k c k d n - k (1)

Signal Cyclic Convolution

Given ft f t and gt g t periodic with the same period T T.

Figure 1
fig1.png
yt=1T∫0Tfτgt-τdτ↔ e n = c n d n y t 1 T τ 0 T f τ g t τ ↔ e n c n d n

notation:

yt=ft*gt y t f t g t
Figure 2
fig2.png
Why is this called circular/periodic convolution?

Proof

e n =1T2∫0T∫0Tfτgt-τdτⅇ-ⅈ ω o ntdt=1T∫0Tfτ1T∫0Tgt-τⅇ-ⅈ ω o ntdtdτ e n 1 T 2 t 0 T τ 0 T f τ g t τ ω o n t 1 T τ 0 T f τ 1 T t 0 T g t τ ω o n t (2)
Where v=t-τ v t τ and ⅆv=ⅆt ⅆ v ⅆ t 1T∫0Tgt-τⅇ-ⅈ ω o ntdt→1T∫-τT-τgvⅇ-ⅈ ω o nv+τdv→1T∫-τT-τgvⅇ-ⅈ ω o nvdvⅇ-ⅈ ω o nτ→ d n ⅇ-ⅈ ω o nτ 1 T t 0 T g t τ ω o n t 1 T v τ T τ g v ω o n v τ 1 T v τ T τ g v ω o n v ω o n τ d n ω o n τ
e n = d n 1T∫0Tfτⅇ-ⅈ ω o nτdτ= c n d n e n d n 1 T τ 0 T f τ ω o n τ c n d n (3)

Square Pulse

Exercise 1

Figure 3
fig3.png
What signal has FS coefficients e n = c n 2=14sin2π2nπ2n2 e n c n 2 1 4 2 n 2 2 n 2

Solution 1

Cyclic convolution of ft f t with itself.

Figure 4
Figure 4 (rain.png)

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