Digital Signal Processing - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 57

The definition of discrete time windows often varies in a subtle way, particularly
for the windows becoming zero at 0 and N - 1, such as the Hanning window. We
thus lose the information brought by the first and last points. We can correct this
oversight by expanding the window to take them into account and increase the
resolution by a factor

N + 2

. For example, the Hanning window is written in some

N

topics as:

⎛ k + 1 ⎞

()

wk = 0.50 cos 2 π

⎠

⎜ ⎟

0, N − 1

N + 1

⎝

2.4. Examples of application

2.4.1. LTI systems identification

The Fourier transform is of particular practical interest for the representation of
dynamic systems. A system consists of a cause-effect relationship between a batch
of input signals and a batch of output signals, indicated by the diagram in Figure
2.10.

Figure 2.10. Symbolic representation of a system

For example, when we take a bath, we use a system with 2 inputs (angular
position of hot and cold water taps) and 2 outputs (flow rate and water temperature).

Here, we will limit ourselves to single input and single output systems. Consider
a system S of input u and output y ; we will provisionally note the cause-effect
relation linking the signals y and u in the form:

y = S ( u )

We call the impulse response of a system S the Dirac delta function response
S (δ) for continuous time systems, or the Kronecker sequence response S (δ) for the
discrete time systems.