4.2 Discrete-time systems  (Page 5/5)

or

We have the remarkable result that for an input of $x\left(n\right)=z^n$ , we get an output of exactly the same form but multiplied by a constant that dependson $z$ and this constant is the z-transform of the impulse response of the system. In other words, if the system is thought of as a matrix oroperator, $z^n$ is analogous to an eigenvector of the system and $H\left(z\right)$ is analogous to the corresponding eigenvalue.

We also know from the properties of the z-transform that convolution in the $n$ domain corresponds to multiplication in the $z$ domain. This means that the z-transforms of $x\left(n\right)$ and $y\left(n\right)$ are related by the simple equation

The z-transform decomposes $x\left(n\right)$ into its various components along $z^n$ which passing through the system simply multiplies that value time $H\left(z\right)$ and the inverse z-transform recombines the components to give the output. This explains why the z-transform is such a powerful operation in lineardiscrete-time system theory. Its kernel is the eigenvector of these systems.

The z-transform of the impulse response of a system is called its transfer function (it transfers the input to the output) and multiplying it timesthe z-transform of the input gives the z-transform of the output for any system and signal where there is a common region of convergence for thetransforms.

Frequency response of discrete-time systems

The frequency response of a Discrete-Time system is something experimentally measurable and something that is a complete description ofa linear, time-invariant system in the same way that the impulse response is. The frequency response of a linear, time-invariant system is definedas the magnitude and phase of the sinusoidal output of the system with a sinusoidal input. More precisely, if

and the output of the system is expressed as

where $T\left(n\right)$ contains no components at $\omega$ , then $M\left(\omega \right)$ is called the magnitude frequency response and $\phi \left(\omega \right)$ is called the phase frequency response. If the system is causal, linear,time-invariant, and stable, $T\left(n\right)$ will approach zero as $n\to \infty$ and the only output will be the pure sinusoid at the same frequency as the input. This is because a sinusoid is a special case of $z^n$ and, therefore, an eigenvector.

If $z$ is a complex variable of the special form

then using Euler's relation of $e^{jx}=cos\left(x\right)+jsin\left(x\right)$ , one has

and therefore, the sinusoidal input of (3.22) is simply the real part of $z^n$ for a particular value of $z$ , and, therefore, the output being sinusoidal is no surprise.

Fundamental theorem of linear, time-invariant systems

The fundamental theorem of calculus states that an integral defined as an inverse derivative and one defined as an area under a curveare the same. The fundamental theorem of algebra states that a polynomial given as a sum of weighted powers of the independentvariable and as a product of first factors of the zeros are the same. The fundamental theorem of arithmetic states that an integerexpressed as a sum of weighted units, tens, hundreds, etc. or as the product of its prime factors is the same.

These fundamental theorems all state equivalences of different ways of expressing or calculating something. The fundamental theorem oflinear, time-invariant systems states calculating the output of a system can be done with the impulse response by convolution or withthe frequency response (or z-transform) with transforms. Stated another way, it says the frequency response can be found fromdirectly calculating the output from a sinusoidal input or by evaluating the z-transform on the unit circle.