Z-transform

In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation.
It can be considered as a discrete-time equivalent of the Laplace transform. This similarity is explored in the theory of time-scale calculus.
History.
The basic idea now known as the Z-transform was known to Laplace, and it was re-introduced in 1947 by W. Hurewicz and others as a way to treat sampled-data control systems used with radar. It gives a tractable way to solve linear, constant-coefficient difference equations. It was later dubbed "the z-transform" by Ragazzini and Zadeh in the sampled-data control group at Columbia University in 1952.
The modified or advanced Z-transform was later developed and popularized by E. I. Jury.
The idea contained within the Z-transform is also known in mathematical literature as the method of generating functions which can be traced back as early as 1730 when it was introduced by de Moivre in conjunction with probability theory.
From a mathematical view the Z-transform can also be viewed as a Laurent series where one views the sequence of numbers under consideration as the (Laurent) expansion of an analytic function.
Definition.
The Z-transform can be defined as either a "one-sided" or "two-sided" transform.
Bilateral Z-transform.
The "bilateral" or "two-sided" Z-transform of a discrete-time signal formula_1 is the formal power series formula_2 defined as
where formula_3 is an integer and formula_4 is, in general, a complex number:
where formula_6 is the magnitude of formula_4, formula_8 is the imaginary unit, and formula_9 is the "complex argument" (also referred to as "angle" or "phase") in radians.
Unilateral Z-transform.
Alternatively, in cases where formula_1 is defined only for formula_11, the "single-sided" or "unilateral" Z-transform is defined as
In signal processing, this definition can be used to evaluate the Z-transform of the unit impulse response of a discrete-time causal system.
An important example of the unilateral Z-transform is the probability-generating function, where the component formula_1 is the probability that a discrete random variable takes the value formula_3, and the function formula_2 is usually written as formula_15 in terms of formula_16. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.
Inverse Z-transform.
The "inverse" Z-transform is
where "C" is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). In the case where the ROC is causal (see Example 2), this means the path "C" must encircle all of the poles of formula_2.
A special case of this contour integral occurs when "C" is the unit circle. This contour can be used when the ROC includes the unit circle, which is always guaranteed when formula_2 is stable, that is, when all the poles are inside the unit circle. With this contour, the inverse Z-transform simplifies to the inverse discrete-time Fourier transform, or Fourier series, of the periodic values of the Z-transform around the unit circle:
The Z-transform with a finite range of "n" and a finite number of uniformly spaced "z" values can be computed efficiently via Bluestein's FFT algorithm. The discrete-time Fourier transform (DTFT)—not to be confused with the discrete Fourier transform (DFT)—is a special case of such a Z-transform obtained by restricting "z" to lie on the unit circle.
Region of convergence.
The region of convergence (ROC) is the set of points in the complex plane for which the Z-transform summation converges.
Example 1 (no ROC).
Let "x[n]" = (0.5)"n". Expanding "x[n]" on the interval (−∞, ∞) it becomes
Looking at the sum
Therefore, there are no values of "z" that satisfy this condition.
Example 2 (causal ROC).
Let formula_23 (where "u" is the Heaviside step function). Expanding "x[n]" on the interval (−∞, ∞) it becomes
Looking at the sum
The last equality arises from the infinite geometric series and the equality only holds if |0.5"z"−1| < 1 which can be rewritten in terms of "z" as |"z"| > 0.5. Thus, the ROC is |"z"| > 0.5. In this case the ROC is the complex plane with a disc of radius 0.5 at the origin "punched out".
Example 3 (anti causal ROC).
Let formula_26 (where "u" is the Heaviside step function). Expanding "x[n]" on the interval (−∞, ∞) it becomes
Looking at the sum
Using the infinite geometric series, again, the equality only holds if |0.5−1"z"| < 1 which can be rewritten in terms of "z" as |"z"| < 0.5. Thus, the ROC is |"z"| < 0.5. In this case the ROC is a disc centered at the origin and of radius 0.5.
What differentiates this example from the previous example is "only" the ROC. This is intentional to demonstrate that the transform result alone is insufficient.
Examples conclusion.
Examples 2 & 3 clearly show that the Z-transform "X(z)" of "x[n]" is unique when and only when specifying the ROC. Creating the pole–zero plot for the causal and anticausal case show that the ROC for either case does not include the pole that is at 0.5. This extends to cases with multiple poles: the ROC will "never" contain poles.
In example 2, the causal system yields an ROC that includes |"z"| = ∞ while the anticausal system in example 3 yields an ROC that includes |"z"| = 0.
In systems with multiple poles it is possible to have a ROC that includes neither |"z"| = ∞ nor |"z"| = 0. The ROC creates a circular band. For example,
has poles at 0.5 and 0.75. The ROC will be 0.5 < |"z"| < 0.75, which includes neither the origin nor infinity. Such a system is called a mixed-causality system as it contains a causal term (0.5)"n""u"["n"] and an anticausal term −(0.75)"n""u"[−"n"−1].
The stability of a system can also be determined by knowing the ROC alone. If the ROC contains the unit circle (i.e., |"z"| = 1) then the system is stable. In the above systems the causal system (Example 2) is stable because |"z"| > 0.5 contains the unit circle.
Let us assume we are provided a Z-transform of a system without a ROC (i.e., an ambiguous "x[n]"). We can determine a unique "x[n]" provided we desire the following:
For stability the ROC must contain the unit circle. If we need a causal system then the ROC must contain infinity and the system function will be a right-sided sequence. If we need an anticausal system then the ROC must contain the origin and the system function will be a left-sided sequence. If we need both stability and causality, all the poles of the system function must be inside the unit circle.
The unique "x[n]" can then be found.
Properties.
Parseval's theorem
Initial value theorem: If "x"["n"] is causal, then
Final value theorem: If the poles of ("z"−1)"X"("z") are inside the unit circle, then
Table of common Z-transform pairs.
Here:
is the unit (or Heaviside) step function and
is the discrete-time unit impulse function (cf Dirac delta function which is a continuous-time version). The two functions are chosen together so that the unit step function is the accumulation (running total) of the unit impulse function.
Relationship to Fourier series and Fourier transform.
For values of formula_4 in the region formula_36, known as the unit circle, we can express the transform as a function of a single, real variable, ω, by defining formula_37. And the bi-lateral transform reduces to a Fourier series:
which is also known as the discrete-time Fourier transform (DTFT) of the formula_1 sequence. This 2-periodic function is the periodic summation of a Fourier transform, which makes it a widely used analysis tool. To understand this, let formula_39 be the Fourier transform of any function, formula_40, whose samples at some interval, "T", equal the x["n"] sequence. Then the DTFT of the "x"["n"] sequence can be written as follows.
 = \frac{1}{T}\sum_{k=-\infty}^{\infty} X(f-k/T).</math>
When "T" has units of seconds, formula_41 has units of hertz. Comparison of the two series reveals that  formula_42  is a normalized frequency with units of "radians per sample". The value ω=2 corresponds to formula_43 Hz.  And now, with the substitution  formula_44  can be expressed in terms of the Fourier transform, X(•):
As parameter T changes, the individual terms of move farther apart or closer together along the f-axis. In however, the centers remain 2 apart, while their widths expand or contract. When sequence "x"("nT") represents the impulse response of an LTI system, these functions are also known as its frequency response. When the formula_45 sequence is periodic, its DTFT is divergent at one or more harmonic frequencies, and zero at all other frequencies. This is often represented by the use of amplitude-variant Dirac delta functions at the harmonic frequencies. Due to periodicity, there are only a finite number of unique amplitudes, which are readily computed by the much simpler discrete Fourier transform (DFT).  (See .)
Relationship to Laplace transform.
Bilinear transform.
The bilinear transform can be used to convert continuous-time filters (represented in the Laplace domain) into discrete-time filters (represented in the Z-domain), and vice versa. The following substitution is used:
to convert some function formula_47 in the Laplace domain to a function formula_48 in the Z-domain (Tustin transformation), or
from the Z-domain to the Laplace domain. Through the bilinear transformation, the complex s-plane (of the Laplace transform) is mapped to the complex z-plane (of the z-transform). While this mapping is (necessarily) nonlinear, it is useful in that it maps the entire formula_50 axis of the s-plane onto the unit circle in the z-plane. As such, the Fourier transform (which is the Laplace transform evaluated on the formula_50 axis) becomes the discrete-time Fourier transform. This assumes that the Fourier transform exists; i.e., that the formula_50 axis is in the region of convergence of the Laplace transform.
Starred transform.
Given a one-sided Z-transform, X(z), of a time-sampled function, the corresponding starred transform produces a Laplace transform and restores the dependence on sampling parameter, T:
The inverse Laplace transform is a mathematical abstraction known as an "impulse-sampled" function.
Linear constant-coefficient difference equation.
The linear constant-coefficient difference (LCCD) equation is a representation for a linear system based on the
autoregressive moving-average equation.
Both sides of the above equation can be divided by α0, if it is not zero, normalizing α0 = 1 and the LCCD equation can be written
This form of the LCCD equation is favorable to make it more explicit that the "current" output "y[n]" is a function of past outputs "y[n−p]", current input "x[n]", and previous inputs "x[n−q]".
Transfer function.
Taking the Z-transform of the above equation (using linearity and time-shifting laws) yields
and rearranging results in
Zeros and poles.
From the fundamental theorem of algebra the numerator has "M" roots (corresponding to zeros of H) and the denominator has N roots (corresponding to poles). Rewriting the transfer function in terms of zeros and poles
where "qk" is the "k"-th zero and "pk" is the "k"-th pole. The zeros and poles are commonly complex and when plotted on the complex plane (z-plane) it is called the pole–zero plot.
In addition, there may also exist zeros and poles at "z" = 0 and "z" = ∞. If we take these poles and zeros as well as multiple-order zeros and poles into consideration, the number of zeros and poles are always equal.
By factoring the denominator, partial fraction decomposition can be used, which can then be transformed back to the time domain. Doing so would result in the impulse response and the linear constant coefficient difference equation of the system.
Output response.
If such a system "H(z)" is driven by a signal "X(z)" then the output is "Y(z)" = "H(z)X(z)". By performing partial fraction decomposition on "Y(z)" and then taking the inverse Z-transform the output "y[n]" can be found. In practice, it is often useful to fractionally decompose formula_59 before multiplying that quantity by "z" to generate a form of "Y(z)" which has terms with easily computable inverse Z-transforms.