The impulse invariant transform (IIT) is a method of taking a continuous-time system H(s) and converting it to a discrete-time system. There are multiple ways of doing this, but the IIT does so with the constraint that the impulse response of the discrete-time system is a sampled version of the impulse response of the continuous-time system.<\/p>\n
Here\u2019s an illustration:<\/p>\n
\u2026gets converted to\u2026<\/p>\n This doesn\u2019t seem like a big deal\u2014nor very accurate. The illustration above implies that we\u2019re taking an IIR response from the continuous-time system and sampling as a discrete-time FIR.<\/p>\n However, the IIT actually does something better: if H(s) is rational (composed of a numerator and denominator):<\/p>\n $$H(s) = \\frac{N_c(s)}{D_c(s)}$$<\/p>\n \u2026then the IIT lets us re-write this system as a discrete-time rational system:<\/p>\n $$H[z] = \\frac{N_d[z]}{D_d[z]}$$<\/p>\n \u2026of course, with the property that $h[n] = T_s \\times h(nT_s)$. $T_s$ is the sampling period (typically, during transformation from continous-time to discrete-time, the continuous-time impulse response is scaled by the sample period).<\/p>\n Here\u2019s an example:<\/p>\n $$H(s) = \\frac{s}{s^2 + 1}$$<\/p>\n First, we break it down into its partial fraction expansion:<\/p>\n $$H(s) = \\frac{s}{(s-j)(s+j)} = \\frac{a}{s-j} + \\frac{b}{s+j}$$<\/p>\n Solving (taking limits of s to +j and -j):<\/p>\n $$a = \\frac{1}{2}$$ Now, the IIT prescribes how to take single-pole transfer functions and convert them to the z-domain.<\/p>\n $$\\frac{a}{s-j} \\rightarrow \\frac{T_s a}{1-e^{+jT_s}z^{-1}}$$ \u2026which means that:<\/p>\n $$H[z] = \\frac{T_s}{2(1-e^{+jT_s}z^{-1})} – \\frac{T_s}{2(1-e^{-jT_s}z^{-1})}$$ The impulse invariant transform is useful in modelling<\/em> continuous-time sigma-delta, allowing one to analyze the mixed-mode continuous-time sigma-delta as a purely discrete-time system.<\/p>\n![\"\"](\"https:\/\/www.circuitdesign.info\/blog\/wp-content\/uploads\/2009\/02\/analogsystemdot.png\")
<\/a><\/p>\n<\/a>
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\n\u2026with the characteristic that the discrete-time impulse response is a sampled version of the continuous-time:<\/p>\n <\/a><\/p>\n
\n<\/a>Rational Systems<\/h2>\n
\n<\/a>Example<\/h2>\n
\n$$b = \\frac{-1}{2}$$<\/p>\n
\n$$\\frac{b}{s+j} \\rightarrow \\frac{T_s b}{1-e^{-jT_s}z^{-1}}$$<\/p>\n
\n$$H[z] = \\frac{T_s cos(jT_s) z^{-1}}{1 – cos(jT_s)z^{-1} + z^{-2}}$$<\/p>\n
\n<\/a>Application<\/h2>\n