# Phase Noise Theory: Ideal Frequency Multipliers and Dividers

Multiplying the frequency of a signal by a factor of N using an ideal frequency multiplier increases the phase noise of the multiplied signal by 20log(N) dB. Similarly dividing a signal frequency by N reduces the phase noise of the output signal by 20log(N) dB.

## Frequency Multipliers

For the signal

f(t) = cos(ωt + θ)

Frequency multiplication by N multiplies the cosine function argument (ωt + θ) by N.

Consequently any phase noise term in the θ is also multiplied by N. (the phase noise is increased by 20log(N) dB).

### Frequency Multiplier with a Frequency Modulated input

For a frequency modulated signal

f(t) = cos (ω_{c}t + β sin(ω_{m}t))

for small β

f(t) = cos ( ω_{c}t) + (β/2)[ cos( ω_{c} - ω_{m})t - cos( ω_{c} + ω_{m})t]

becomes after frequency multiplication by N

f(t) = cos(Nω_{c}t + Nβ sin(ω_{c}t))

for small Nβ

f(t) = cos(Nω_{c}t) + (Nβ/2)[cos(Nω_{c} - ω_{m})t - cos(Nω_{c} + ω_{m})t]

### NOTES

1) The sideband amplitude is increased by N. i.e by 20Log(N) dB.

2) The sideband offset from the carrier in the frequency multiplied signal is the same as for the original signal.

## Frequency Dividers

For the signal

f(t) = cos(ωt + θ)

Frequency division by N divides the cosine function argument (ωt + θ) by N.

Consequently any phase noise term in the θ is also divided by N. (the phase noise is decreased by 20log(N) dB).

### Frequency Divider with a Frequency Modulated input

For a frequency modulated signal

f(t) = cos(ω_{c}t + βsin(ω_{m}t))

for small β

f(t) = cos(ω_{c}t) + (β/2)[cos(ω_{c} - ω_{m})t - cos(ω_{c} + ω_{m})t]

becomes after frequency division by N

f(t) = cos((ω_{c}/N)t + (β/N)sin(ω_{c}t))

for small β/N

f(t) = cos((ω_{c}/N)t) + (β/2N)[cos(ω_{c}/N - ω_{m})t - cos(ω_{c}/N + ω_{m})t]

### NOTES

1) The sideband amplitude is decreased by N. i.e by 20Log(N) dB.

2) The sideband offset from the carrier in the frequency divided signal is the same as for the original signal.

### Derivations for particular multipliers and dividers

For derivations of the above results for particular frequency multipliers and dividers see:

Derivations using Complex numbers

Derivations using Trigonometric Identities

### REFERENCES

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