Frequency modulation
Frequency modulation
Frequency modulation (FM) is an electronic communication/transmission method by altering the carrier signal's frequency to match the data signal, while the carrier's strength (amplitude) is unchanged.
Embedded Files
[1]
Note: Unlike AM, FM uses rules, formulas, and values not present in the process of AM, like Carson's bandwidth rule, frequency deviation. Both do use concepts like modulation index but via different formulas and symbols.
Components
Components
The carrier frequency (fC) is the max shift in a carrier frequency away from its unmodulated center/nominal frequency when a modulating signal is applied, mathematically defined as Δf = fmβ.
The frequency deviation (Δf) is the max shift in a carrier frequency away from its unmodulated center/nominal frequency when a modulating signal is applied, mathematically defined as Δf = fmβ.
The modulation frequency (fm) is the message signal's frequency, mathematically defined as fm = Δf/β.
The modulation index (β) , mathematically defined as β = Δf/fm.
Note: Not confusing FM modulation index symbol 'β' with 'm' for AM modulation index symbol.
Carson's bandwidth rule
Carson's bandwidth rule
Carson's bandwidth rule is an FM formula that calculates the near-total power bandwidth of a FM signal and the total sidebands that could fit into it.
It states that the bandwidth is twice the sum of the peak frequency deviation and the highest modulating frequency.
The (standard) formula of Carson's bandwidth rule formula is: BW = 2(Δf + fm).
Δf = peak frequency deviation
fm = modulation frequency
There is also a second Carson's bandwidth formula using the modulation index: BW = 2fm(β + 1).
To derive this formula, we substitute Δf = fmβ into the standard Carson's rule formula 2(Δf + fm): 2(fmβ + fm) → 2fm(β + 1).
[Q1]
[Q1] In this spectrum analyzer screen signal, the modulation frequency can be determined by the fact that the peaks are roughly 0.1 kHz apart from each other
The frequency deviation can be determined by... TBA
But in this case, both values are also indicated on the screen window as 100 Hz modulation and 240.9 Hz deviation.
The modulation index is β = 240.9Hz/100Hz = 2.41
The bandwidth, according to Carson's rule, is: BW = 2*(2.41Hz + 100Hz) = 204.82
Calculating the number of significant sidebands in a bandwidth We can also determine the number of (significant) sidebands in the measurement:
By adding 1 to the index accounts for the carrier component and ensures we capture enough sideband energy to have 98% the total signal power: 2.41+1 = 3.41, so Carson's rule indicates there are 3 sideband pairs, so 3x2 = 6 sidebands can fit in the measurement above.
Or straighforwardly, the total number of sidebands (not pairs) in the bandwidth is calculated as: n = 2(Δf/fm + 1) = 2(β + 1) (Carson's bandwidth rule but without fm), where "Δf/fm + 1" calculates the number of sidebands (on only 1 side).
Although ensure to multiply the number of sidebands as a whole number, so this process shouldnt be done directly with this formula.
Calculating the total power of significant sidebands in a bandwidth The total power of all sidebands that fall in Carson's bandwidth is: (10-8.6/10 + 10-16.5/10 + 10-9.3/10) x 2 = 0.556 mW.
Calculating the total power of unsignificant sidebands in a bandwidth The total power of all sidebands outside Carson's bandwidth is (4th, 5th, 6th peaks' powers): (10-26.3/10 + 10-38/10 + 10-52/10) x 2 = 0.005 mW.
[Q1]
E.g,. in this spectrum analyzer indicates that 240.9Hz/100Hz + 1 = 3 sidebands fall within the Carson's bandwidth (do not use 3.4 sidebands).
So the total number of sidebands falling into the sidebands is also 3x2 = 6.
The total power of the sidebands in the bandwidth is: 10*log(.083mW) + 10*log(.015mW) + 10*log(.077mW) = -40 dBm; or
(.083mW + .015mW + .077mW)x2 = 0.35 mW.
The total power of the within the sidebands (including the carrier frequency: 10-57.6/10 = 0.0000017378 mW) is still 0.922 mW for the carrier being so low.
Percentage of the total power within Carson's bandwidth
Percentage of the total power within Carson's bandwidth
The percentage of the total power within Carson's bandwidth defines the percentage of the total power within Carson's bandwidth (all significant sidebands and the carrier frequency's power sum) over the total power over the whole signal. Its value is often within 98-99%
Its formula is % = PBW/PT.
To calculate the total power's percentage falling within Carson's bandwidth rule, we must also calculate the total power of sidebands outside Carson's bandwidth and the entire signal:
The total power of sidebands outside Carson's bandwidth is: (0.0015mW+0.00015mW)x2 = 0.0033 mW
The total power of the whole signal is: 0.0033mW + 0.35 mW = 0.3533 mW.
So the total percentage of the total power within Carson's bandwidth is: PBW/PT = 0.35mW/0.3533mW = 99%
Bessel chart
Bessel chart
The Bessel chart lists the mathematical values of Bessel functions of the first kind (Jn), or in this case how an FM signal's total power/amplitude is distributed between the center carrier frequency and its surrounding sidebands. It's used in FM to find the relative amplitudes and number of significant sidebands for a given modulation.
[Q1]
In this Bessel chart, the values under the 1st, 2nd, 3rd sidebands are their scaling factors, which are used to be multiplied by the unmodulated carrier amplitude.
The dashes meant the value is 0.
E.g., Given Δf = fm = 100 Hz, then the index is 100/100 = 1.
Locating the row, the corresponding carrier of this is 0.77.
The carrier voltage formula is Vn = Jn x VC. So the 1st sideband's amplitude is 141mV*0.44 = 63.36 mV.
Sideband frequency calculation E.g., if an FM carrier signal has a 10 kHz frequency and a 141 mV amplitude. The FM frequency is 200 Hz and the deviation frequency is 400 Hz, we can calculate the sidebands frequencies, like the 3rd USB:
Via the formula fUSBn = fC + (n x fm) = 10kHz + (3*200Hz) = 10.6 kHz
Why does sidebands occur alongside modulated carrier when message and carrier signals are mixed? - carrier and message signals are mathematically trig functions (like cos: cos(ωct) and cos(ωmt)), which are multiplied by modulator. The product-to-sum identity in trigonometry dictates what occurs when 2 cos multiply:
ωc = carrier freq
ωm = message freq
And if ωc and ωm are subtituted into the identity/equation, the multiplication is replaced with a sum of 2 entirely new freqs:
References
References
[1] Wikipedia
[1.1] Carson bandwidth rule
[2]
Quizzes
Quizzes
[Q1]
Page updated
Google Sites
Report abuse