# Bruce's: 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

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

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