Relationship to .CE.94-modulation Delta-sigma modulation



fig. 2: derivation of ΔΣ- Δ-modulation


ΔΣ modulation (sdm) inspired Δ modulation (dm), shown in fig. 2. if quantization homogeneous (e.g., if linear), following sufficient derivation of equivalence of dm , sdm:



however, quantizer not homogeneous, , explanation flawed. s true ΔΣ inspired Δ-modulation, 2 distinct in operation. first block diagram in fig. 2, integrator in feedback path can removed if feedback taken directly input of low-pass filter. hence, delta modulation of input signal




u



{\displaystyle \scriptstyle u}

, low-pass filter sees signal








y

dm


=

quantize


(
u


y

dm


)

.



{\displaystyle y_{\text{dm}}=\int \operatorname {quantize} \left(u-y_{\text{dm}}\right).\,}



however, sigma-delta modulation of same input signal places @ low-pass filter








y

sdm


=
quantize


(


(
u


y

sdm


)

)

.



{\displaystyle y_{\text{sdm}}=\operatorname {quantize} \left(\int \left(u-y_{\text{sdm}}\right)\right).\,}



in other words, sdm , dm swap position of integrator , quantizer. net effect simpler implementation has added benefit of shaping quantization noise away signals of interest (i.e., signals of interest low-pass filtered while quantization noise high-pass filtered). effect becomes more dramatic increased oversampling, allows quantization noise programmable. on other hand, Δ-modulation shapes both noise , signal equally.


additionally, quantizer (e.g., comparator) used in dm has small output representing small step , down quantized approximation of input while quantizer used in sdm must take values outside of range of input signal, shown in fig. 3.



fig. 3: example of sdm of 100 samples of 1 period sine wave. 1-bit samples (e.g., comparator output) overlaid sine wave logic high (e.g.,




+

v

c
c





{\displaystyle \scriptstyle +v_{cc}}

) represented blue , logic low (e.g.,






v

c
c





{\displaystyle \scriptstyle -v_{cc}}

) represented white.



in general, ΔΣ has advantages versus Δ modulation:



the whole structure simpler:

only 1 integrator needed
the demodulator can simple linear filter (e.g., rc or lc filter) reconstruct signal
the quantizer (e.g., comparator) can have full-scale outputs


the quantized value integral of difference signal, makes less sensitive rate of change of signal.






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