Note: In practice, it is not possible to generate Dirac deltas for
the creation of
xs
t
xs
t
. So, instead of convolving with
δt
δt
, we might convolve with a rectangular
pulse of width TT, known as a
zero-order hold (ZOH) function and denoted by
hzt
hz
t
. This yields
xzt
xz
t
:

xzt=∑n
=−∞∞xnhz
t−nT
xz
t
n
xn
hz
t
n
T

(1)
Unwanted spectral copies in

Xz
iΩ
Xz
Ω
can be removed by a final stage of lowpass filtering.
The two-step procedure is illustrated in

Figure 1. As with the Dirac delta, we assume a ZOH
function with unit area, requiring that the analog
reconstruction lowpass filter has DC gain

T
T.

Unlike convolution with
δt
δt
, convolution with the ZOH function
hz
t
hz
t
introduces passband "droop" and out-of-band
attenuation. This results from the fact that the frequency
response magnitude of the ZOH function is not constant in
Ω:

H
z
iω=∫−∞∞
hz
te−(iΩt)d
t
=∫0T1Te−(iΩt)dt
=sinΩT2ΩT2e−(iΩT2)
H
z
ω
t
hz
t
Ω
t
t
0
T
1
T
Ω
t
Ω
T
2
Ω
T
2
Ω
T
2

(2)
(See

Figure 2 for an illustration.) Thus, for perfect
reconstruction, the analog reconstruction filter

Hr
iΩ
Hr
Ω
must invert the ZOH response

Hz
iΩ
Hz
Ω
over the passband

−πT
πT
T
T
.
The left side of

Figure 3 shows the ideal

|Hr
iΩ|
Hr
Ω
.

It is difficult to build analog reconstruction filters with sharp
cutoff. Instead, one hopes that the desired signal is
bandlimited to less than the Nyquist frequency (as in Figure 1),
so that there is an absence of unwanted spectral energy in a
region around
±πT
±
T
. In this case, the analog reconstruction filter
Hr
iΩ
Hr
Ω
may be designed with a wider transition band, such as on the
right side of Figure 3.