Discrete Time Fourier Transform Pair Discrepancy

Question

When taking the DTFT of a discrete-time signal $x[n]=a \times u[n]$, the book "Signals and Systems" by Chi-Tsong Chen claims that the result is $\frac 1 {1-a \cdot e^{-j \cdot \omega \cdot T}}$; however, many other sources claim the DTFT is $\frac 1 {1-a \cdot e^{-j \cdot \omega}}$ without the T term.

The book I have states that $x[n]=x(nT)$ by definition and the DTFT by definition is

$$X(\omega)= \sum_{n=-\infty}^{+\infty}x(nT)e^{-j \omega n T}$$.

Can someone explain the reason for the differences?

Thank you for your time.

Answer 1

This is just normalized frequency versus un-normalized frequency.

Check the capitalization of omega. The capital omega $\Omega$, usually refers to the normalized frequency (with $-\pi$ and $\pi$ corresponding to $-f_s/2$ and $f_s/2$) and the normal omega $\omega$, to the regular angular frequency.

Your two sources just use different convention. There is no contradiction.