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CTFS

Module by: Richard Baraniuk

Summary: Ths module is an overview of the CTFS.

xt x t is a periodic CT signal with period T T

Figure 1
fig1.png
or equivalently finite-length CT signal of length T T
Figure 2
fig2.png

CTFS Coefficients

c k =1T∫0Txtⅇ-ⅈ2πTktdt c k 1 T t 0 T x t 2 T k t (1)
where -∞<k<∞ k

Reconstruction

xt=∑k=-∞∞ c k ⅇⅈ2πTkt x t k c k 2 T k t (2)

Exercise 1

Find the CTFS as limit of DFT as N→∞ N

Exercise 2

CTFS as projection onto any normal basis of sinusoids b k t=ⅇⅈ 2 π TktT b k t 2 T k t T where 0≤t<T 0 t T , and -∞<k<∞ k

Exercise 3

CTFS ≡ DTFT with time and frequency variables interchanged.

DTFT CTFS
n k
ω t

In the sequel, change of notation:

signal xt→ft x t f t

FS coefficients c k → c n c k c n

2πT= ω o 2 T ω o

ft=∑n∈z c n ⅇⅈ2πTnt f t n n z c n 2 T n t (3)
where c n =1T∫0Tftⅇ-ⅈ2πTntdt c n 1 T t 0 T f t 2 T n t

Recall:

Figure 3
fig3.png
The frequency of a sinusoid =# of cycles2πlength interval=# of cycles2πT # of cycles 2 length interval # of cycles 2 T Define ω o =2πT ω o 2 T to be the fundamental frequency

Then the basis functions are ⅇ ⅈ ω o n t n∈ℤ ω o n t n ℤ

Note:

All frequencies of ⅇ ⅈ ω o n t ω o n t are n ω o n ω o , that is, integer multiples of ω o ω o
Obviously as T T increases, ω o ω o dereases.
Figure 4
fig4.png
ft=∑x∈ℤ c n ⅇⅈ ω o nt f t n x ℤ c n ω o n t (4)
c n =1T∫0Tftⅇ-ⅈ ω o ntdt c n 1 T t 0 T f t ω o n t (5)
Figure 5
fig5.png

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