# Period of a sum of cosine signals

• Quincy
In summary, the period of a sum of cosine signals can be calculated by finding the least common multiple (LCM) of the individual periods of each signal, using the formula T = 2π/ω. The sum can have a longer period than the individual signals, and the amplitude of the individual signals does not affect the period. If one signal has a negative amplitude, the period will not change. Additionally, the sum will always have a repeating period, as the individual signals have periodic behavior.
Quincy

## Homework Statement

Find the period of the signal x(t) = 2 + 4cos(40Pi*t) + 3cos(60Pi*t) + 4cos(120Pi*t).

## The Attempt at a Solution

The fundamental frequency (fo) = 10 Hz, since that's the greatest common factor of all the frequencies of the cosine signals. So the period would be 1/10 = 0.1 s. But my professor took points off for it, why?

how many times per sec does each component of the signal by itself repeat?

How much time does each component of the signal take to go through one cycle?

## 1. What is the formula for calculating the period of a sum of cosine signals?

The period of a sum of cosine signals can be calculated by finding the least common multiple (LCM) of the individual periods of each cosine signal in the sum. This can be represented by the formula T = 2π/ω, where ω is the angular frequency.

## 2. Can the period of a sum of cosine signals be longer than the individual periods?

Yes, the period of a sum of cosine signals can be longer than the individual periods. This is because the LCM of two or more numbers is always greater than or equal to the individual numbers.

## 3. How does the amplitude of the individual cosine signals affect the period of the sum?

The amplitude of the individual cosine signals does not affect the period of the sum. The period is only affected by the frequency or angular frequency of the individual signals.

## 4. What happens to the period if one of the cosine signals has a negative amplitude?

If one of the cosine signals has a negative amplitude, the period of the sum will not change. Negative amplitude only affects the phase and not the frequency or period of the signal.

## 5. Is it possible for the sum of cosine signals to have a non-repeating period?

No, the sum of cosine signals will always have a repeating period. This is because the individual cosine signals have periodic behavior and the sum of these signals will also exhibit periodic behavior.

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