Sum of two continuous periodic signals is always periodic
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A real function is said to be periodic if there exists a real number so that for all . The number is said to be a period of the function. The most familiar examples of periodic functions are the trigonometric functions sine, cosine, and tangent.
Note that if is a period of a function, then are also periods. If a periodic function is continuous and nonconstant, then it has a least period, and all other periods are positive integer multiples of the least period.
There are discontinuous periodic functions that have no least period. The most famous example is the Dirichlet function, defined by if is rational, if is irrational. Every positive rational number is a period of the Dirichlet function, so there is no least period.
Sums of periodic functions are often periodic
The sum of two periodic functions is often periodic, but not always. Suppose that is periodic with least period , and is periodic with least period. If and have a common multiple, then is periodic with period. However,is not necessarily the least period of. To give an extreme example,has no least period.
Note that if is a period of a function, then are also periods. If a periodic function is continuous and nonconstant, then it has a least period, and all other periods are positive integer multiples of the least period.
There are discontinuous periodic functions that have no least period. The most famous example is the Dirichlet function, defined by if is rational, if is irrational. Every positive rational number is a period of the Dirichlet function, so there is no least period.
Sums of periodic functions are often periodic
The sum of two periodic functions is often periodic, but not always. Suppose that is periodic with least period , and is periodic with least period. If and have a common multiple, then is periodic with period. However,is not necessarily the least period of. To give an extreme example,has no least period.
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