Digital to Analog Conversion

1
Digital to Analog Conversion
Sometimes the output of a DSP system can be used
directly, without conversion of a digital signal
back to an analog signal. This is often the case
in signal analysis systems, and sometimes in
control or communication systems. However, in
many cases the DSP system must interface with the
analog world at both ends (input and output). In
those cases, the digital signal must be converted
back to an analog form. The digital signal is a
sequence of integers in binary format. The first
step in converting the signal from digital to
analog form is to convert each integer in the
sequence to a Voltage (or current). There are
many circuits which do this, and a discussion of
their internal operation is beyond the scope of
this course.
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Digital to Analog Conversion
For the sake of this discussion, well call this
first step in the digital to analog conversion
process a binary to level conversion conversion
of an integer in binary format to a Voltage or
current level. If the input to the
binary-to-level converter was an analog Voltage
signal, and if the binary-to-level converter
produces a Voltage level, the output Voltage
level may be thought of as an estimate of the
input signal Voltage
which produced the nth sample in the sequence.
Binary to Level Conversion
V(n)
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Digital to Analog Conversion
The binary-to-level converter is followed by a
zero-order hold circuit. This is a sort of
reverse sample-and-hold. It takes one sample (as
a level, not as an integer), and holds that level
for one sample period. The output of the
zero-order hold is a sort of staircase
approximation to the analog signal well
eventually produce.
Binary to Level Conversion
Zero-Order Hold
V(n)
4
Digital to Analog Conversion
The binary-to-level converter is followed by a
zero-order hold circuit. This is a sort of
reverse sample-and-hold. It takes one sample (as
a level, not as an integer), and holds that level
for one sample period. The output of the
zero-order hold is a sort of staircase
approximation to the analog signal well
eventually produce.
Binary to Level Conversion
Zero-Order Hold
V(n)
5
Digital to Analog Conversion
Heres a 1 kHz sine wave, before sampling.
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Digital to Analog Conversion
Heres the same 1 kHz sine wave, with samples
taken at fs 22 kHz.
7
Digital to Analog Conversion
Heres the sequence of samples. For a
high-resolution system, this would be a good
approximation to the input to the zero-order hold.
The Matlab stem command was used to plot the
sample values in this fashion.
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Digital to Analog Conversion
And here are the samples along with the staircase
output of the zero-order hold.
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Digital to Analog Conversion
The output of the zero-order hold is still, in
effect, a discrete time signal. It only changes
at integer multiples of the sample period, so it
still has the spectral characteristics of a
discrete time signal. This means it still has an
image at every integer multiple of the sample
frequency.
Power Spectral density
B
-B
fs
2fs
-fs
fs/2
-fs/2
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Digital to Analog Conversion
The last step in recovery of the analog signal is
to remove the images with an anti-imaging, or
recovery, filter. This filter could have the
same frequency response as the anti-aliasing
filter at the other end of the system.
Power Spectral density
Frequency response of recovery filter
B
-B
fs
2fs
-fs
fs/2
-fs/2
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Digital to Analog Conversion
Heres the time-domain view of the zero-order
holds output. If we smooth it, itll be a good
approximation to the original sine wave.
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Digital to Analog Conversion
We add the recovery filter previously mentioned
to strip off the spectral images. This is a
lowpass filter, so it smooths the staircase
output. Removing the images is the
frequency-domain view of the effect of the
recovery filter, smoothing the stairsteps is the
time domain view of the same action.
Binary to Level Conversion
Zero-Order Hold
Recovery Filter
V(n)
V(t)
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Digital to Analog Conversion
Heres are the staircase output from the
zero-order hold, along with the smoothed output
of the recovery filter. This completes the
digital-to-analog conversion.
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Digital to Analog Conversion
These three stages, the binary-to-level
conversion, zero-order hold, and recovery filter,
comprise a digital-to-analog converter, or DAC.
Binary to Level Conversion
Zero-Order Hold
Recovery Filter
V(n)
V(t)
Digital-to-Analog Converter (DAC)
15
Analog-to-Digital Conversion
At the input end of the system, we have the
analog-to-digital converter, or ADC. The ADC may
inclode both the quantizer and the
sample-and-hold, or it may consist of the
quantizer only, with an external sample-and-hold.
Sample And Hold
N-bit quantizer
V(t)
Vq(n)
Analog-to-Digital Converter (ADC)