# FSK: Selecting center frequencies and Carson's Rule

I'm implementing 2FSK and 4FSK modulation schemes for an experimental system. I'm using GNURadio, although that shouldn't affect the question.

I'm trying to figure out how the center frequency should be selected. Many 2FSK examples use a center frequency of 1700Hz, with a deviation of 500Hz, for a "space" frequency of 1200Hz and a "mark" frequency of 2200Hz.

1. How are these values arrived at, though? From my understanding, only the deviation affects the modulation index and the occupied bandwidth (Carson's rule). So, why use 1700Hz +/- 500Hz, instead of 1000Hz +/- 500Hz?

2. Is there a lower bound? i.e. would it be possible to use a center frequency of 500Hz, for a "space" at 0Hz (DC) and a "mark" at 1000Hz?

3. I'd also like to confirm how deviation is defined in 4FSK. In 2FSK, the symbol values are at fc - fd and fc + fd, where fc = center frequency and fd = frequency deviation from fc. In 4FSK, is it:

• Symbol values are at fc - 3fd, fc - fd, fc + fd, fc + 3fd, or
• Symbol values are at fc - 2fd/3, fc - fd/3, fc + fd/3, fc + 2fd/3

i.e. is fd the deviation from center to the first symbol value, or from center to the last (outermost) symbol value?

I'm trying to determine how to estimate the occupied bandwidth of a 4FSK signal using Carson's rule: BW = 2(fd + fb) where fb = bitrate, but it's unclear what the value of fd would be here in 4FSK.

1. As a concrete example: what would the occupied bandwidth be for a 4FSK signal at 50 baud (100 bits/second), where the center frequency is 2000Hz and the symbol values are at 1250Hz, 1750Hz, 2250Hz, and 2750Hz?

1. You're working purely in the digital domain, but most amateur FSK modes are designed to be compatible with analog systems, where the AF has a limited bandwidth, maybe something like 300 through 3000 Hz. This makes it a good idea to use a center frequency somewhere around 1500 through 1700 Hz because it will be right in the middle of the filter passband, and subject to the least attenutation or distortion. But it's not an absolute requirement.

2. With your all-digital chain, you can probably go down to 0Hz, but if you're working with real samples you don't want to go below 0Hz or you'll have aliasing problems (100Hz and -100Hz will be indistinguishable).

3 and 4. In my experience, MFSK systems usually talk about "tone spacing" rather than "deviation", where the tone spacing is the difference between two adjacent symbol frequencies. But Carson's rule says peak deviation, which you can consider to be the difference between the center frequency and the furthest symbol frequency from the center — in your example, the peak deviation is 750Hz, and the Carson bandwidth is 2 * (750 + 50) = 1600 Hz. A simpler way to arrive at the same answer would be to subtract the lowest from the highest symbol frequency to arrive at 1500Hz, and then add double the modulation rate.

Of course, you should remember that Carson is only an estimate; "pure" FSK is modulating with a square wave and therefore has effectively infinite sidebands (although most of the power will be within the Carson bandwidth); GFSK, on the other hand, will be significantly narrower.

Carson's rule doesn't apply to (low-order) FSK; it makes a statement about frequency-modulated continuous-spectrum signals, not about signals that jump (pulse-shaped) between a small set of frequencies.