The discrete-time Fourier transform (DTFT) of real or complex numbers $\{x_n:n\in\mathbb Z\}$ is a Fourier series, which produces a periodic function of a frequency variable and is given by $$ X(\omega) = \sum_{n=-\infty}^{\infty} x_n \,e^{-i \omega n}, $$ where $i$ is the imaginary unit.

I am interested in the differentiability of the DTFT. Is it a differentiable function? What conditions on $\{x_n:n\in\mathbb Z\}$ are needed for the DTFT to be differentiable? Is there an expression for the derivative? Can the series be differentiated term by term?

In particular, I am interested in the differentiability of the spectral density function $f:[-\pi,\pi]\to[0,\infty)$ defined by $$ f(\lambda)=\frac1{2\pi}\sum_{t=-\infty}^\infty\gamma_t e^{-i\lambda t}, $$ where $\{\gamma_t:t\in\mathbb Z\}$ are absolutely summable autocovariances of some weakly stationary random variables. Is $f$ a differentiable function?

Any help is much appreciated!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.