This document explains Digital-to-Analog Conversion (DAC), the process of reconstructing an analog signal from discrete-time samples. It begins with ideal reconstruction using an ideal low-pass filter (LPF) in both frequency and time domains, using sinc interpolation to reconstruct the continuous signal. Practical DAC implementations are then explored, introducing components like zero-order hold circuits and non-ideal reconstruction filters. The frequency response of the zero-order hold is discussed, along with the need for a compensated filter to correct magnitude distortions. The document emphasizes the challenges of real-world reconstruction and illustrates the DAC process in both time and frequency domains for clarity.

1. Topic 5
Digital to Analog Conversion
1

2. Introduction
 According to the Nyquist Sampling
Theorem, if the analog signal, xa(t), is
bandlimited, then the signal can be
recovered from the samples if the
sampling frequency used is at least twice
the highest frequency component in xa(t).
 This is done by using an ideal lowpass
filter called a reconstruction filter.
2

3. Frequency domain analysis of
ideal reconstruction
 Recall what happens in the frequency
domain when an analog signal, xa(t) gets
sampled to give xs(t):
◦ Xs(j), consists of repeated copies Xa(j),
scaled by 1/Ts and separated in frequency by
s=2/Ts:
3
)
)
(
(
1
)
( 








k
s
a
s
s k
j
X
T
j
X

4. Frequency domain analysis of
ideal reconstruction
 If aliasing does not occur, then xa(t) can
be recovered exactly from xs(t) using an
ideal lowpass filter(LPF).
 An ideal LPF has a box function freq.
response:
 The cutoff frequency, c is usually chosen
to be s/2=/Ts.
4


 




otherwise
T
j
H c
s
r
,
0
|
|
,
)
(
)
( 
j
Hr

)
2
/
(
/ s
s
T 


s
T
/


s
T

5. Frequency domain analysis of
ideal reconstruction
 The filter has a gain of Ts since Xs(j) is
the scaled version of Xa(j) by 1/Ts.
 The reconstruction process can be
represented in the frequency domain as:
i.e. multiplication in the freq.domain.
 The reconstructed signal has a FT, Xr(j),
that is exactly equal to Xa(j). (See
Appendix I of Topic 4)
5
)
(
)
(
)
( 


 j
H
j
X
j
X r
s
r

6. Time domain analysis of ideal
reconstruction
 The inverse FT of a box function is a sinc
function.
 For the ideal LPF with the frequency
response:
the inverse FT (or impulse response) is:
 The frequency response Hr(j), and impulse
response, hr(t), are shown in Appendix I.
6


 



otherwise
T
T
j
H s
s
r
,
0
/
|
|
,
)
(

 
t
t
t
h
s
s
T
T
r 

sin
)
( 

7. Time domain analysis of ideal
reconstruction
 In the time domain, the reconstructed
signal is obtained from the convolution:
 This gives:
7






n
s
r
r nT
t
h
n
x
t
x )
(
]
[
)
(



 


n s
s
s
s
r
T
nT
t
T
nT
t
n
x
t
x
/
)
(
)
/
)
(
sin(
]
[
)
(



8. Time domain analysis of ideal
reconstruction
 So what does the equation mean?
 The reconstructed signal, xr(t), is obtained
by adding together shifted versions (by nTs)
of the sinc function that have been scaled
by the sample value, x[n].
 The ideal LPF interpolates between the
impulses of xs(t) to construct xr(t).
 This is depicted in the figure in Appendix
I.
8

9. Practical D-to-A Converters
 In the real-world we cannot achieve ideal
D/A conversion.
◦ Ideal LPFs with sharp cutoff (box function) do
not exist.
◦ Conversion is not instantaneous.
9

10. Practical D-to-A Converters
 A practical D-to-A converter has the
following elements:
◦ Zero-order hold – After the sequence of
samples are converted to impulses, this system
holds the quantized samples for one sample
period. The impulse response is:
◦ Reconstruction (smoothing) filter – A non-
ideal LPF with a cutoff of around s/2=/Ts.
10


 


otherwise
T
t
t
h s
,
0
0
,
1
)
(
0

11. Practical D-to-A Converters
 The figure below shows the outputs at
different stages of the DAC.
11







n
s
s
a
a
a
s nT
t
nT
y
t
s
t
y
t
y )
(
)
(
)
(
)
(
)
( 
t
s
T
2
 s
T
 0 s
T s
T
2 s
T
3 s
T
4 s
T
5 t
s
T
2
 s
T
 0 s
T s
T
2 s
T
3 s
T
4 s
T
5
t
)
(
]
[ s
a nT
y
n
y 
n
2
 1
 0 1 2 3 4 5
Ts
1
0
t
h0(t)
convert to
impulses
zero-order
hold
]
[n
y )
(t
ys )
(
'
t
ya
)
(t
ya
Compensated
reconstruction
filter

12. Practical D-to-A Converters
 The frequency response of the zero-order hold is:
 A comparison of the frequency response of the
zero-order hold and interpolator filter can be
found in Appendix II.
 The zero-order hold does not have a sharp cutoff
like the ideal interpolator filter.
 Therefore, some frequency components above
s/2=/Ts will remain at the output of the zero-
order hold.
12
  2
/
2
/
sin
2
)
(
0
s
T
j
e
T
j
H s 






13. Practical D-to-A Converters
 Finally, the output of the zero-order hold
goes through a reconstruction filter.
 To compensate for the “droop” or
magnitude distortion in the frequency
response of the zero-order hold we use a
compensated reconstruction filter, which
compensates the magnitude response.
 An ideal compensated reconstruction filter
will have a frequency response as shown
in Appendix II.
13

14. Practical D-to-A Converters
 The following figure shows the DAC process in
the frequency domain.
14
t
s
T
2
 s
T
 0 s
T s
T
2 s
T
3 s
T
4 s
T
5
t
)
(
]
[ s
a nT
y
n
y 
n
2
 1
 0 1 2 3 4 5
convert to
impulses
zero-order
hold
]
[n
y )
(t
ys )
(
'
t
ya
)
(t
ya
Compensated
reconstruction
filter







n
s
s
a
a
a
s nT
t
nT
y
t
s
t
y
t
y )
(
)
(
)
(
)
(
)
( 
t
s
T
2
 s
T
 0 s
T s
T
2 s
T
3 s
T
4 s
T
5
... ...













k
s
a
T
a
a
s jk
j
Y
j
S
j
Y
j
Y s
)
(
)
(
)
(
)
( 1
2
1


0
s
T
1
s

2
s

 2 s

s

 0

0


s
s T

2


..
.
...

0
s
T
1
s

2
s

 2 s

s

 0

0


s
s T

2


..
.
...

0
s
T
1
0

0


s
T



After compensation