Calculating Group Delay in Delta Sigma Converters
Delta-sigma converter group delay equals the phase derivative across decimation filter stages, calculated as K times M minus one divided by twice the modulator clock for sinc filters.
Transfer Function Z Domain
Mathematical Model
Mathematical representation of a discrete-time linear system relates the output sequence to the input sequence using complex frequency variables. For digital signal processing components, the transfer function z domain provides an algebraic method to analyze the filter response and phase characteristics. This representation consists of a ratio of polynomials in the complex variable z.
System Stability
System response is analyzed by finding the roots of these polynomials to locate the poles and zeros of the filter. The positions of these roots on the complex plane dictate the stability and frequency response of the system. If any poles lie outside the unit circle, the digital system becomes unstable and will oscillate.
Calibration Alignment
Metrological calibration utilizes this mathematical model to predict and correct the frequency response of digitized sensor systems. By fitting the measured frequency response to a theoretical transfer function z domain, engineers can extract the exact filter parameters and coefficients. This alignment process ensures that the digital output corresponds accurately to the physical input.
Practical Verification
Experimental verification involves driving the digital filter with known test sequences and analyzing the output to extract the coefficients of the transfer function. When the hardware implements fixed-point arithmetic, coefficient quantization can shift the poles and zeros, which requires a comparison between the ideal model and the actual measured response. The results of this verification are documented to certify the accuracy and stability of the measurement system before it is deployed in high-reliability applications.