What is z signal transformation. Inverse z-transform

In the analysis and synthesis of discrete and digital devices, the so-called z-transforms are widely used, which play the same role with respect to discrete signals as the Fourier and Laplace integral transform with respect to continuous signals.

Definition of z-transform

Let be a numerical sequence, finite or infinite, containing the reference values ​​of some signal. We put it in a one-to-one correspondence with the sum of a series in negative powers of the complex variable z:

Let's call this sum, if it exists, the z-transform of the sequence . The expediency of introducing such a mathematical object is related to the fact that the properties of discrete sequences of numbers can be studied by studying their z-transforms using the usual methods of mathematical analysis. In mathematics, the z-transform is also called the generating function of the original sequence.

Based on formula (1.46), one can directly find the z-transforms of discrete signals with a finite number of samples. So, the simplest discrete signal with a single sample corresponds to . If, for example, then

Series Convergence

If the number of terms in the series (1.46) is infinitely large, then it is necessary to investigate its convergence. The following is known from the theory of functions of a complex variable. Let the coefficients of the considered series satisfy the condition

for any . Here and are constant real numbers.

Then series (1.46) converges for all values ​​of z such that . In this region of convergence, the sum of the series is an analytic function of the variable z, which has neither poles nor essentially singular points.

Let us consider, for example, a discrete signal formed by the same single samples and serving as a model of the usual switching function. Endless row

is the sum of a geometric progression and converges for any z in the annulus. Summing up the progression, we get:

At the boundary of the domain of analyticity for z = 1 this function has a single simple pole. Similarly, the z-transform of an infinite discrete signal is obtained, where a is some real number. Here:

This expression makes sense in some annular region.

Z-transform of continuous functions

Assuming that the readings are the values ​​of a continuous function at points , any signal can be associated with its z-transform at the selected sampling step:

For example, if , then the corresponding z-transform

is an analytic function for .

Inverse z-transform

Let p-content/image_post/osncifr/pic45_8.gif> be a function of a complex variable z analytic in the annular region . A remarkable property of the z-transform is that the function determines the entire infinite collection of samples. Indeed, we multiply both parts of the series (1.46) by the factor :

Then we calculate the integrals from both parts of the resulting equality, taking as an integration contour an arbitrary closed curve that lies entirely in the analyticity region and covers all poles:

The bypass of the integration contour is carried out in the positive direction, counterclockwise.

To solve equation (1.50), we use the fundamental position following from the Cauchy theorem:

Obviously, the integrals of all terms on the right side of expression (1.50) will vanish, except for the term with the number m, That's why

Formula (1.51) is called the inverse z-transform.

Example

A z-transform of the form is given. Find the coefficients of the discrete signal corresponding to this function.

First of all, we define that the function is analytic in the entire plane, except for the point , so it can indeed be the z-transform of some discrete signal.

Before solving this problem, let's recall from the course of higher mathematics the technique for solving curvilinear integrals using the theory of residues and the Cauchy residue theorem. Let the point be an isolated singular point of the function . The residue of a function at a point is a number denoted by the symbol and defined by the equality:

As a contour g, we can take a circle centered at a point of sufficiently small radius such that the circle does not go beyond the analyticity region of the function

And it did not contain functions inside other special points. The residue of the function is equal to the coefficient of minus the first power in the Laurent expansion in the vicinity of the point : . The residue at a removable singular point is equal to zero.

If a point is a pole n th order of the function , then

In the case of a simple pole ()

If a function in a neighborhood of a point can be represented as a quotient of two analytic functions

and , i.e. is a simple pole of the function , then

Turning to formula (1.48), we find that

for any idth=41 height=19 src=http://electrono.ru/wp-content/image_post/osncifr/pic46_12.gif> . Thus, the original discrete signal has the form:

Relation to the Laplace and Fourier transforms

Let us define for a signal of the form of an ideal MIP:

Transforming it according to Laplace, we get an image for any constant a and b. This property can be proved by substituting the sum into formula (1.46). is a sequence of numbers whose common term is equal to:

Such a discrete convolution, in contrast to a circular convolution, is sometimes called a linear convolution.

Let's calculate the z-transform of the discrete convolution:

The convolution of two discrete signals corresponds to the product of their z-transforms.

In the analysis and synthesis of discrete and digital devices, the so-called z-transform is widely used, which plays the same role with respect to discrete signals as the integral Fourier and Laplace transforms with respect to continuous signals. This section outlines the foundations of the theory of this functional transformation and some of its properties.

Definition z -conversions. Let be a numerical sequence, finite or infinite, containing the reference values ​​of some signal. We put it in a one-to-one correspondence with the sum of a series in negative powers of a complex variable z:

Let's call this sum, if it exists, z-transformation sequences (X to }. The expediency of introducing such a mathematical object is related to the fact that the properties of discrete sequences of numbers can be studied by studying their z-transforms using the usual methods of mathematical analysis.

Based on formula (2.113), one can directly find the z-transforms of discrete signals with a finite number of samples. So, the simplest discrete signal with unity corresponds to the exact reading.

If, for example,

Series convergence. If the number of terms in the series (2.113) is infinitely large, then it is necessary to investigate its convergence. The following is known from the theory of functions of a complex variable. Let the coefficients of the considered series satisfy the condition

for any . Here M > 0 and R 0 > 0 - constant real numbers. Then the series (2.113) converges for all values ​​of z such that |z| > R 0 . In this region of convergence, the sum of the series is an analytic function of the variable z, which has neither poles nor essentially singular points.

Let us consider, for example, a discrete signal formed by the same single samples and serving as a model of the usual switching function. The infinite series is the sum of a geometric progression and converges for any z in the ring.

Summing up the progression, we get

At the boundary of the domain of analyticity for z= 1, this function has a single simple pole.

Similarly, the z-transform of an infinite discrete signal is obtained, where a - some real number. Here

This expression makes sense in the annular region.

z -conversion of continuous functions. Assuming that the readings are the values ​​of a continuous function x(t) at points , any signal x(t) you can map it to the z-transform at the chosen sampling rate:

For example, if , then the corresponding z-transform

.

is an analytic function for .

Reversez-conversion. Let be X(z) is a function of the complex variable z, analytic in the annular region |z| > R 0 . A remarkable property of the z-transform is that the function X(z) defines the entire infinite collection of samples .

Indeed, we multiply both parts of the series (2.113) by the factor :

. (2.115)

and then we calculate the integrals of both parts of the resulting equality, taking as an integration contour an arbitrary closed curve that lies entirely in the analyticity region and envelops all the poles of the function X(z). In this case, we use the –fundamental position following from the Cauchy theorem:

.

Obviously, the integrals of all terms on the right-hand side will vanish, with the exception of the term with the number t, That's why

This formula is called reverse z -transformation .

Connection with Laplace and Fourier transforms . Let us define for a signal of the form of an ideal MIP:

.

Transforming it according to Laplace, we get the image

which goes directly to the z-transform if we perform the substitution . If we put , then the expression

Let's return to the discrete Fourier transform formula:

In the theory of discrete systems, it is customary to use a slightly different form of notation associated with the introduction of a Z-transformation. Let's make the following substitution:

.

Then the above formula is greatly simplified:

.

The newly obtained function X(z) of the variable z is called the Z-image or Z-image of the discrete signal x(k).

Z-transforms for discrete signals and systems play the same role as the Laplace transform for analog systems. Therefore, let's consider a number of examples of determining Z - images of some typical discrete signals.

1.Single impulse(Fig. 9.14) is a discrete analogue of the δ - impulse and is a single report with a single value:

Z - the transformation of a single impulse is found as

as for the δ - Dirac momentum.

2. Discrete single jump(Fig. 9.15) is a complete analogue of the Heaviside inclusion function:

Z - the image of a unit jump can be found as

The resulting sum is the sum of the terms of an infinite geometric progression with the initial term equal to 1 and the denominator
. The sum of the terms of the series is:

.

3. Discrete exponent(Fig. 9.16) is a signal defined by the expression:

At
the discrete exponent is decreasing (Fig. 9.16), when
- increasing, at
- alternating.Z - the image of such an exponent

As in the previous case, we got a geometric progression with a zero term equal to one, but with a denominator
. The infinite sum of the terms of the progression determines the Z - the image of the exponent:

4. Discrete damped harmonic. In contrast to the previous examples, we write it in general form:

G de α – harmonic damping factor,

ω is the harmonic frequency,

φ is the initial phase of oscillations,

- sampling period.

Let us introduce the following notation:

Figure 9.17 shows a graph of the discrete damped harmonic with the following data: a=0.9,
, φ=π/9. Taking into account the accepted notation, the expression for the discrete damped harmonic can be represented as:

.

When obtaining the Z-image of the harmonic, the cosine function should be expressed through the sum of two complex exponents. Then, having done a number of algebraic and trigonometric transformations, in the end, it will be possible to obtain the following expression:

.

From the above examples, it can be seen that Z - images of most discrete signals are fractional rational functions of the variable
. The origin of the Z-transform from the Laplace and Fourier transforms leads to the fact that the Z-transform has similar properties.

1. Linearity.

Z - the transformation is linear, so if there are two signals , then the sum of these signals
has a Z-image
.

2. Discrete signal time delay.

If a discrete signal x(k) having Z is the image of X(z), delay by m sampling steps
, then the delayed signal y(k)=x(k-m) has Z - the image
. Expression
can be thought of as a signal delay operator by one sampling step.

3. Convolution of discrete signals.

By analogy with convolution of analog signals

,

Fourier - the image of which is equal to the product of the Fourier - images of convoluted signals, the convolution of two discrete signals is defined as

.

Z - the image of the convolution of two signals is equal to the product of Z - images of the original discrete signals

4. Multiplication by discrete exponent.

If the discrete signal
, having a Z-image
, multiplied by the exponent
, then Z - the image of the product will take the form
.

The considered properties of the Z-transformation make it possible in many cases to easily find the Z-image of a given signal or solve the inverse problem - using the known Z-image of the signal to find its representation in time.

The Z-transform is mainly used to calculate discrete filters. The mathematical apparatus of the z-transform plays the same role for digital devices as for analog circuits. Using the z-transform, frequency filters, phase correctors or Hilbert transforms are easily calculated for their implementation in digital form. Let's immediately separate the concepts of discrete and digital filters. In discrete filters, the impulse response is discrete in time, but the signal samples and filter parameters can take on any value. In digital filters, both signal samples and filter parameters (for example, coefficients) are represented by binary numbers of a certain capacity. An example of a discrete filter is a switched capacitor filter.

When considering the sampling of signals, we found that the spectrum of the input analog signal, when converted to a discrete form, repeats along the frequency axis an infinite number of times. The same thing happens with the frequency response of a discrete filter. An example of a change in the amplitude-frequency characteristic of a low-pass filter during its discrete implementation is shown in Figure 1.

Figure 1. An example of the frequency response of a discrete filter

In the example shown, the sample rate is 50 kHz. Therefore, two more passbands of the discrete filter are formed near this frequency. A discrete filter, such as a switched capacitor filter or a digital filter, will require an analog anti-aliasing filter to work properly to suppress the high frequency components of the input signal. Its idealized frequency response is shown in red in Figure 1.

If there is an analog filter transfer characteristic H(s) in the form of zeros and poles of the filter, then in the discrete filter zeros and poles are periodically repeated with period 1/ T, where T is the sampling period. In other words, the filter is repeated in this way as shown in Figure 1. The position of zeros and poles on the s-plane frequency axis for conventional and discrete filters is shown in Figure 2.

Figure 2. Periodic repetition of zeros and poles on the s-plane

At the discrete filter, we see an infinite number of zeros and poles, which is not very convenient for its implementation. Instead of endless repetition of zeros and poles on an infinite frequency axis, you can convert this axis to a ring one (use the polar coordinate system instead of the Cartesian one). Such a transformation is shown in Figure 3.

Figure 3. Converting a complex s-plane to a complex z-plane

In this transformation, the zero frequency occupies the position of the +1 point on the real z-plane axis, the frequency equal to ∞ is transformed to the −1 point on the real z-plane axis, and the frequency axis itself is transformed into a circle of unit radius. As the frequency increases, we will move in a circle counterclockwise, thereby realizing the infinite repetition of the amplitude-frequency characteristics of the discrete filter.

Mathematically, the mapping from the complex s-plane to the complex z-plane is done as follows:

Z = e s T (1)

where s = σ + jω

Then the Laplace transform of the discrete signal goes into the z-transform:

(2)

When passing from the complex s-plane to the complex z-plane, all infinitely repeating zeros and poles of a discrete filter in the s-plane are mapped to a finite number of zeros and poles in the z-plane. Then the expression for the transfer characteristic of the discrete filter can be represented as follows:

(3)