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Transform Analysis of LTI Systems using Z-Transform
Z-Transform
The Z-transform is a mathematical tool which is used to convert the difference equations in discrete time domain into the algebraic equations in z-domain. Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete time function, then its Z-transform is defined as,
$$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$
Transform Analysis of Discrete-Time System
The Z-transform plays a vital role in the design and analysis of discrete-time LTI (Linear Time Invariant) systems.
Transfer Function of a Discrete-Time LTI System
The figure shows a discrete-time LTI system having an impulse response $\mathit{h}\mathrm{\left(\mathit{n}\right)}$.
Consider the system gives an output $\mathit{y}\mathrm{\left(\mathit{n}\right)}$ for an input $\mathit{x}\mathrm{\left(\mathit{n}\right)}$. Then,
$$\mathrm{\mathit{y}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\mathit{h}\mathrm{\left(\mathit{n}\right)}*\mathit{x}\mathrm{\left(\mathit{n}\right)}}$$
Taking Z-transform on both the sides, we get,
$$\mathrm{\mathit{Z}\mathrm{\left[ \mathit{y}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{\mathrm{=}}\mathit{Z}\mathrm{\left[\mathit{h}\mathrm{\left(\mathit{n}\right)}*\mathit{x}\mathrm{\left(\mathit{n}\right)} \right ]}}$$
$$\mathrm{\therefore \mathit{Y}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\mathit{H}\mathrm{\left(\mathit{z}\right)}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$
Therefore, the Z-transform of the impulse response $\mathit{h}\mathrm{\left(\mathit{n}\right)}$ of the system is given by,
$$\mathrm{\mathit{H}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\frac{\mathit{Y}\mathrm{\left(\mathit{z}\right)}}{\mathit{X}\mathrm{\left(\mathit{z}\right)}}}$$
Where, H(z) is called the transfer function of the discrete-time LTI system and can be defined as follows −
The transfer function of a discrete time LTI system is defined as the ratio of Z-transform of the output sequence to the Z-transform of the input sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$, when the initial conditions are neglected.
Relationship between Transfer Function and Difference Equation of Discrete Time LTI System
An nth order discrete-time LTI system is described in terms of a difference equation as follows −
$$\mathrm{\sum_{\mathit{k=\mathrm{0}}}^{\mathit{N}}\mathit{a_{\mathit{k}}}\mathit{y}\mathrm{\left(\mathit{n-k}\right)}\:\mathrm{=}\:\sum_{\mathit{k=\mathrm{0}}}^{\mathit{M}}\mathit{b_{\mathit{k}}}\mathit{x}\mathrm{\left(\mathit{n-k}\right)}}$$
On expanding the above difference equation, we get,
$$\mathrm{\mathit{a_{\mathrm{0}}}\mathit{y}\mathrm{\left(\mathit{n}\right)}\:\mathrm{+}\:\mathit{a_{\mathrm{1}}}\mathit{y}\mathrm{\left(\mathit{n-\mathrm{1}}\right)}\:\mathrm{+}\:\mathit{a_{\mathrm{2}}}\mathit{y}\mathrm{\left(\mathit{n-\mathrm{2}}\right)}\:\mathrm{+}\:\mathit{a_{\mathrm{3}}}\mathit{y}\mathrm{\left(\mathit{n-\mathrm{3}}\right)}\:\mathrm{+}\:...\:\mathrm{+}\:\mathit{a_{\mathit{N}}}\mathit{y}\mathrm{\left(\mathit{n-\mathit{N}}\right)}\:\mathrm{=}\:\mathit{b_{\mathrm{0}}}\mathit{x}\mathrm{\left(\mathit{n}\right)}\:\mathrm{+}\:\mathit{b_{\mathrm{1}}}\mathit{x}\mathrm{\left(\mathit{n-\mathrm{1}}\right)}\:\mathrm{+}\:\mathit{b_{\mathrm{2}}}\mathit{x}\mathrm{\left(\mathit{n-\mathrm{2}}\right)}\:\mathrm{+}\:\mathit{b_{\mathrm{3}}}\mathit{x}\mathrm{\left(\mathit{n-\mathrm{3}}\right)}\:\mathrm{+}\:...\:\mathrm{+}\:\mathit{b_{\mathit{M}}}\mathit{x}\mathrm{\left(\mathit{n-\mathit{M}}\right)}}$$
Taking Z-transform on both sides and neglecting the initial conditions, we get,
$$\mathrm{\mathit{Z}\mathrm{\left [ \mathit{a_{\mathrm{0}}}\mathit{y}\mathrm{\left(\mathit{n}\right)}\:\mathrm{+}\:\mathit{a_{\mathrm{1}}}\mathit{y}\mathrm{\left(\mathit{n-\mathrm{1}}\right)}\:\mathrm{+}\:\mathit{a_{\mathrm{2}}}\mathit{y}\mathrm{\left(\mathit{n-\mathrm{2}}\right)}\:\mathrm{+}\:\mathit{a_{\mathrm{3}}}\mathit{y}\mathrm{\left(\mathit{n-\mathrm{3}}\right)}\:\mathrm{+}\:...\:\mathrm{+}\:\mathit{a_{\mathit{N}}}\mathit{y}\mathrm{\left(\mathit{n-\mathit{N}}\right)} \right ]}\:\mathrm{=}\:\mathit{Z}\mathrm{\left[\mathit{b_{\mathrm{0}}}\mathit{x}\mathrm{\left(\mathit{n}\right)}\:\mathrm{+}\:\mathit{b_{\mathrm{1}}}\mathit{x}\mathrm{\left(\mathit{n-\mathrm{1}}\right)}\:\mathrm{+}\:\mathit{b_{\mathrm{2}}}\mathit{x}\mathrm{\left(\mathit{n-\mathrm{2}}\right)}\:\mathrm{+}\:\mathit{b_{\mathrm{3}}}\mathit{x}\mathrm{\left(\mathit{n-\mathrm{3}}\right)}\:\mathrm{+}\:...\:\mathrm{+}\:\mathit{b_{\mathit{M}}}\mathit{x}\mathrm{\left(\mathit{n-\mathit{M}}\right)}\right]}}$$
$$\mathrm{\Rightarrow \mathit{a_{\mathrm{0}}}\mathit{Y}\mathrm{\left(\mathit{z}\right)}\:\mathrm{+}\:\mathit{a_{\mathrm{1}}}\mathit{z^{-\mathrm{1}}}\mathit{Y}\mathrm{\left(\mathit{z}\right)}\:\mathrm{+}\:\mathit{a_{\mathrm{2}}}\mathit{z^{-\mathrm{2}}}\mathit{Y}\mathrm{\left(\mathit{z}\right)}\:\mathrm{+}\:\mathit{a_{\mathrm{2}}}\mathit{z^{-\mathrm{3}}}\mathit{Y}\mathrm{\left(\mathit{z}\right)}\:\mathrm{+}\:...\mathrm{+}\mathit{a_{\mathit{N}}}\mathit{z^{-\mathit{N}}}\mathit{Y}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\mathit{b_{\mathrm{0}}}\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{+}\:\mathit{b_{\mathrm{1}}}\mathit{z^{-\mathrm{1}}}\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{+}\:\mathit{b_{\mathrm{2}}}\mathit{z^{-\mathrm{2}}}\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{+}\:\mathit{b_{\mathrm{3}}}\mathit{z^{-\mathrm{3}}}\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{+}\:...\mathrm{+}\mathit{b_{\mathit{M}}}\mathit{z^{-\mathit{M}}}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$
$$\mathrm{\Rightarrow \mathrm{\left[ \mathit{a_{\mathrm{0}}}\:\mathrm{+}\:\mathit{a_{\mathrm{1}}}\mathit{z^{-\mathrm{1}}}\:\mathrm{+}\:\mathit{a_{\mathrm{2}}}\mathit{z^{-\mathrm{2}}}\:\mathrm{+}\:\mathit{a_{\mathrm{3}}}\mathit{z^{-\mathrm{3}}}\:\mathrm{+}\:...\:\mathrm{+}\:\mathit{a_{\mathit{N}}}\mathit{z^{-\mathit{N}}}\right]}\mathit{Y}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\mathrm{\left [ \mathit{b_{\mathrm{0}}}\:\mathrm{+}\:\mathit{b_{\mathrm{1}}}\mathit{z^{-\mathrm{1}}}\:\mathrm{+}\:\mathit{b_{\mathrm{2}}}\mathit{z^{-\mathrm{2}}}\:\mathrm{+}\:\mathit{b_{\mathrm{3}}}\mathit{z^{-\mathrm{3}}}\:\mathrm{+}\:...\:\mathrm{+}\:\mathit{b_{\mathit{M}}}\mathit{z^{-\mathit{M}}} \right ]}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$
$$\mathrm{\Rightarrow\frac{\mathit{Y}\mathrm{\left(\mathit{z}\right)}}{\mathit{X}\mathrm{\left(\mathit{z}\right)}}\:\mathrm{=}\:\frac{\mathit{b_{\mathrm{0}}}\:\mathrm{+}\:\mathit{b_{\mathrm{1}}}\mathit{z^{-\mathrm{1}}}\:\mathrm{+}\:\mathit{b_{\mathrm{2}}}\mathit{z^{-\mathrm{2}}}\:\mathrm{+}\:\mathit{b_{\mathrm{3}}}\mathit{z^{-\mathrm{3}}}\:\mathrm{+}\:...\:\mathrm{+}\:\mathit{b_{\mathit{M}}}\mathit{z^{-\mathit{M}}}}{\mathit{a_{\mathrm{0}}}\:\mathrm{+}\:\mathit{a_{\mathrm{1}}}\mathit{z^{-\mathrm{1}}}\:\mathrm{+}\:\mathit{a_{\mathrm{2}}}\mathit{z^{-\mathrm{2}}}\:\mathrm{+}\:\mathit{a_{\mathrm{3}}}\mathit{z^{-\mathrm{3}}}\:\mathrm{+}\:...\:\mathrm{+}\:\mathit{a_{\mathit{N}}}\mathit{z^{-\mathit{N}}}}}$$
$$\mathrm{\therefore \frac{\mathit{Y}\mathrm{\left(\mathit{z}\right)}}{\mathit{X}\mathrm{\left(\mathit{z}\right)}}\:\mathrm{=}\:\mathit{H}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\frac{\sum_{\mathit{k=\mathrm{0}}}^{\mathit{M}}\mathit{b}_{\mathit{k}}\mathit{z}^{-\mathit{k}}}{\sum_{\mathit{k=\mathrm{0}}}^{\mathit{N}}\mathit{a}_{\mathit{k}}\mathit{z}^{-\mathit{k}}}}$$
Where, $\mathit{H}\mathrm{\left(\mathit{z}\right)}$ is the transfer function of the discrete-time system and the above equation gives the relation between the transfer function and the difference equation of the system.
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