# Non Periodic, Deterministic Power Signals

Any one know of work on non-periodic but deterministic power signals? Now one member in this class would be the quasi periodic signals. I wonder if there is a generalized Fourier analysis of non-periodic deterministic power signals.

• Fourier analysis does apply to non-periodic signals... The spectrum is just continuous rather than discrete. The conditions of existence of the Fourier transform of a signal is that it has to be deterministic and can be integrated on $\mathbb{R}$ or $\mathbb{C}$ – Florent Jul 20 '17 at 1:54
• Also a generalisation of the Fourier transform is the Laplace transform (cf wikipedia) – Florent Jul 20 '17 at 1:57
• I talk about a power signal here. Fourier transforms are only defined for some energy signals. – John Woods Jul 21 '17 at 15:06

Consider the signal $$x(t) = \sum_{n=-\infty}^\infty \operatorname{rect}(t-n)\sin(2\pi nt)$$ which consists of $|n|$ periods of the signal $\sin(2\pi nt)$ in the interval $\left(n-\frac 12,n+\frac 12\right)$ for each integer $n$. Clearly, $x(t)$ is not periodic. Equally clearly, $x(t)$ is a power signal with average power $\frac 12$, and thus it is a nonperiodic but deterministic power signal. I leave it to the OP to determine whether the signal in question has a Fourier transform in the generalized sense (meaning it involves impulses) or not.
• $x(t)$ is even continuous, but the derivatives are not at the $\operatorname{rect}(\cdot)$ boundaries. how 'bout this one? : $$x(t) = \sum_{n=-\infty}^\infty e^{-\pi(t-n)^2} \sin(2\pi nt)$$ – robert bristow-johnson Jul 17 '18 at 2:27