# Raised-Cosine and Root-Raised-Cosine Formulas

### Raised-Cosine (RC) and Root-Raised-Cosine (RRC) filters are commonly used in digital data modems. Here, we provide design formulas for these filters.

#### Notes:

- Each version of the same formula is algebraically equivalent; they just use different parameters.
- The impulse response formulas include a "sinc(x)" term, that is, "sin(x)/x". Although "sinc(0)" seemingly includes a divide-by-zero, L'Hospital's rule indicates that the numeric value of sinc(0) is 1.0. Therefore, whenever you find a sinc(0) condition in using these formulas, just substitute 1.0.

### 1. Raised Cosine Forumlas

### 1.1 Frequency Response

**Version 1**

Equation 9-2-26 of

Digital Communications[Pro95]:Definitions:

fis the frequency.Tis the symbol time.betais the rolloff factor.

**Version 2**

Equation 2.74 of

Digital Communications: Fundamentals and Applications[Skl88]:Definitions:

fis the frequencyW - Wois the excess bandwidth. (The rolloff factor is (W - Wo)/Wo.)

### 1.2 Impuse Response

**Version 1**

Equation 9-2-27 of

Digital Communications[Pro95]:Definitions:

tis the time.Tis the symbol time.betais the rolloff factor.

#### Version 2

The following appears in the book

Digital Communications: Fundamentals and Applications[Skl88]:cos[2pi(W - Wo)t]

h(t) = 2Wo(sinc 2WoT) -------------------

1 - (4(W - Wo)t)^2

( Note: The book is missing some parentheses in denomintor term which have been added here.)

**Version 3**

sin(bt) cos(at)

h(t) = ------- * ------------

bt 1-(2at/pi)^2

where b = 2 pi f

and a = 2 pi excess bandwidth.

### 2. Root-Raised Cosine Formulas

### 2.1 Frequency Response

Since the "root" aspect of a root-raised cosine filter is in the frequency domain, simply take the square root of the raised cosine frequency response given above to get the root-raised cosine frequency response.

### 2.2 Impulse Response

#### Version 1

Contributed by Clay S. Turner:

`The RRC impulse response is given by:`

h(t) = pi^2 4at cos(t(a+b))+pi sin(t(b-a))

----------- * ------------------------------

pi(a-b)-4a t(16t^2 a^2 - pi^2)

where b= 2 pi f (f is usually half of your symbol rate)

and a= 2 pi excess bandwidth

#### Version 2

Contributed by Jim Shima:

`total taps in filter are N (odd)`

Ne = (N-1)/2

s = Ts/T

k = index

then,

for n=0.. N

let k = (n - Ne) (make causal)

h(n) = T * sin[ pi * (r-1) * k * s]

- 4 * r * Ts * k * cos[ pi*(r+1) * k*s] }

--------------------------------------------------------------

T^(0.5) * pi * k * Ts * [ 16*r^2*k^2*s^2 - 1]

0/0 case:

h(n) = T^(-0.5)* [ 1 - r*(1 - (4/pi)) ]

must also take care of any x/0 cases.

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