Oversampling Delta-sigma modulation
fig. 5: noise shaping curves , noise spectrum in ΔΣ modulator
let s consider signal @ frequency
f
0
{\displaystyle \scriptstyle f_{0}}
, sampling frequency of
f
s
{\displaystyle \scriptstyle f_{\mathrm {s} }}
higher nyquist rate (see fig. 5). ΔΣ modulation based on technique of oversampling reduce noise in band of interest (green), avoids use of high-precision analog circuits anti-aliasing filter. quantization noise same both in nyquist converter (in yellow) , in oversampling converter (in blue), distributed on larger spectrum. in ΔΣ-converters, noise further reduced @ low frequencies, band signal of interest is, , increased @ higher frequencies, can filtered. technique known noise shaping.
for first order delta sigma modulator, noise shaped filter transfer function
h
n
(
z
)
=
[
1
−
z
−
1
]
{\displaystyle \scriptstyle h_{n}(z)\,=\,\left[1-z^{-1}\right]}
. assuming sampling frequency
f
s
≫
f
0
{\displaystyle \scriptstyle f_{s}\,\gg \,f_{0}}
, quantization noise in desired signal bandwidth can approximated as:
n
0
=
e
r
m
s
π
3
(
2
f
0
τ
)
3
2
{\displaystyle \mathrm {n_{0}} \,=\,e_{rms}{\frac {\pi }{\sqrt {3}}}\,(2f_{0}\tau )^{\frac {3}{2}}}
.
similarly second order delta sigma modulator, noise shaped filter transfer function
h
n
(
z
)
=
[
1
−
z
−
1
]
2
{\displaystyle \scriptstyle h_{n}(z)\,=\,\left[1-z^{-1}\right]^{2}}
. in-band quantization noise can approximated as:
n
0
=
e
r
m
s
π
2
5
(
2
f
0
τ
)
5
2
{\displaystyle \mathrm {n_{0}} \,=\,e_{rms}{\frac {\pi ^{2}}{\sqrt {5}}}\,(2f_{0}\tau )^{\frac {5}{2}}}
.
in general,
n
{\displaystyle \scriptstyle \mathrm {n} }
-order ΔΣ-modulator, variance of in-band quantization noise:
n
0
=
e
r
m
s
π
n
2
n
+
1
(
2
f
0
τ
)
2
n
+
1
2
{\displaystyle \mathrm {n_{0}} \,=\,e_{rms}{\frac {\pi ^{n}}{\sqrt {2n+1}}}\,(2f_{0}\tau )^{\frac {2n+1}{2}}}
.
when sampling frequency doubled, signal quantization noise improved
10
log
10
(
2
)
(
2
n
+
1
)
d
b
{\displaystyle \scriptstyle 10\log _{10}(2)(2n+1)\,\mathrm {db} }
n
{\displaystyle \scriptstyle \mathrm {n} }
-order ΔΣ-modulator. higher oversampling ratio, higher signal-to-noise ratio , higher resolution in bits.
another key aspect given oversampling speed/resolution tradeoff. in fact, decimation filter put after modulator not filters whole sampled signal in band of interest (cutting noise @ higher frequencies), reduces frequency of signal increasing resolution. obtained sort of averaging of higher data rate bitstream.
example of decimation
let s have, instance, 8:1 decimation filter , 1-bit bitstream; if have input stream 10010110, counting number of ones, 4. decimation result 4/8 = 0.5. can represent 3-bits number 100 (binary), means half of largest possible number. in other words,
the sample frequency reduced factor of eight
the serial (1-bit) input bus becomes parallel (3-bits) output bus.
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