unit for modulation index in FM

[brahma]

i started angle modulation in COMMUNICATION SYSTEM by SIMON HAYKIN in that modulation index β=Δf/fm, where fm=freq of a modulating signal.
Δf is in hertz and fm is in hertz how β got a unit of radians????

 

[flatulent]

It is assigned the units of radians. Radians are not derived from the ratio of the two frequencies.

 

[neils_arm_strong]

β= Δf/fm.
β occurs as the part of phase of the carrier.Since it is a ratio ,it can be given the unit radians.radian does not have any dimensions whatsoever.
Take for example ω=dΘ/dt (angular frequency) its unit is radian ie angle per second
Even if you take the arc of length l of a circle of radius r,then the angle subtended by this lenght at the centre of the circle is Θ=l/r which is in radians even when l/r is a ratio.

 

[Kral]

Brahma,

Neils_Arm_Strong is correct. The radian (or degree) is what I like to refer to as a "label of convenience", rather than a unit. It is awkward to say " an angle equivalent to n radiith of the the way around a circle, so we attach the label "radian". The same thing applies to degrees. It's awkward to say an angle equivalent to 7/360th of the way around the circumference of a circle, so we attach the label "degree" and say 7 degrees".
Regards,
Kral

 

[tamerakshar]

i don't remember that β unit was radians.

 

[tjr]

madulation index is a dimensionless quantity. it has no units

 

[freepoint]

correct, it has no unit, do n ot confused.