The discrete-time Fourier transform, abbreviated DTFT, represents a doubly infinite sequence 
by a periodic function
of a continuous frequency variable. One common convention is
 |
Here 
is the imaginary unit. If 
, this series
has uniform convergence to a continuous
function with period 
. The inverse formula is
 |
Thus the samples 
are Fourier series coefficients
of 
,
with the indicated sign convention.
The frequency variable of the DTFT is continuous, unlike that of the discrete Fourier transform. If 
outside 
, ..., 
, the discrete Fourier
transform with negative exponential and no forward normalization samples the
DTFT at 
for 
,
..., 
.
