DSP Chapter III

Transform as a tool to analyze signals and LTI systems.

The Z-transform is the processing of discrete signals, and Laplace is the processing of continuous signals.

The Z-field is the complex plane of Z.

The Z-transform is just a form of signal expression.

A time series corresponds to X (z) and the Convergent field.

Table 3.1 is a complete summary of the preceding:

Finite time causality, anti-causality, and bilateral signals. Corresponding convergent domain: Remove 0, remove positive infinity, remove 0 and positive infinity.

Infinite time of causality, anti-causality, and bilateral signals. corresponding convergent domain: Outer circle, Circle, ring.

The single-sided Z-transform is prepared for causal signals. The inverse Z-transform formulated.

Table 3.2 is a complete summary of Section 3.2:

Each Z-transform is divided into two pieces, the Z-domain transformation and the convergent domain.

Linear: Easy expression, convergent domain is at least the intersection of the two.

Time Shift: Move-K, multiply z^-k only consider the original effect of the new item

Zoom in the Z field. Easy and easy

Time rollover easy easy is applied to the relevant function in the back

Conjugate easy unchanged

Time domain differentiation: easy and unchanging

Convolution: equivalent to the z-domain multiplication. At least two of the defined domains intersect

Related: Easy and easy

The initial value theorem.

Multiply and Parseval's.

The z-transform of a special function remembers the most special one.

It is impossible to contain any one pole in the convergent domain.

The coefficient of a polynomial is a real coefficient, and its root is the real roots or complex conjugate root. This explains the fact that 3.3.2 is talking about fruit signals.

The extremum point of a single real extremum may have 6 positions. The radius and the 1 ratio determine whether to increase or decrease, in the positive and negative half of the axis determines whether it is a positive or negative.

Two-stage real-value causal signal, because the order changes will be different from unfiltered.

The complex conjugate single-order causal signal is similar to the damping image.

The two-stage expansion begins at a radius of 1.

The system function is the Z-transform of the impulse response. system function has not 0 pole is IIR, only the full 0 system is fir.

Z's inverse transformation has three methods: the first is the contour integral (retention method), the second is the expansion of power series, the third part of the fractional expansion

Contour Integration Method:

The Cauchy integral theorem is first given: C is a closed curve, the F (z) derivative exists in C, and F (z) has no pole at z=z0. The right side of the equals sign is called the number of left (left in curve C). Extended to a more general case, the integrand is seen as f (z)/g (z), and F (z) has no poles within the contour, and G (z) has n discrete first-order poles in C 3.4.4 there is no understanding. There is the opportunity to supplement the basis of the complex function , but it should be concluded in 3.4.6, first back down. example 3.4.1 of the n=-2 of the situation also did not understand ...

Inverse transformation based on series expansion:

Long Division is used directly, but the order of the long divide is determined according to causality and non-causality.

Partial Fractional method: One is to require all poles to be different, the other is complex conjugate poles to produce complex conjugate coefficients.

After a bit of chaos, or the first to do a better problem.

The 3.5 rational Z-transform is a form in which it can be written as a ratio of two polynomial, and most real signals have a rational Z-transform. and the system function because of the pole, the 0-point form numerator denominator is also the polynomial ratio. The Y (z) After multiplying is still the ratio of the polynomial of Z.

Suppose the poles of X (z) and H (z) are different, and there is no 0-point elimination, and all Poles are unfiltered poles. In this way, the inverse Z-transform can be obtained by fractional expansion method. The function of the poles produced by H (Z) is called the natural response of the system, and the coefficient AK in front of each pole is a reflection of the input

The impact of the system. The other part is the forced response, which is the effect of the system on the input.

Because the natural response of the system is to consider the impact of the input on the system, the natural response differs from the 0 input response.

195 the corresponding function of the above multiple poles (which may or may not be caused by either side)

Because it is assumed that the output signal is a causal signal when the inverse Z-transform is made, it can be seen that the 0 State response (initial relaxation), i.e. the response of the psychic state is divided into natural response and forced response. The transition state response means that when n approaches infinity, the value tends to be 0.

To maintain steady-state response, the input must be kept consistent. Example 3.5.1 is to find the pole, 0.5 is definitely the transition state, modulo 1 is the steady state.

LTI is causal, when and only if the convergent domain is on the outer side of the circle (including z= Infinity)

LTI is stable when and only if the convergence field of the system includes the unit circle. More specifically, the causal LTI system is bibo when and only if all Poles are inside the unit circle.

The pole 0 point elimination, is equivalent to numerator. However, the perfect pole 0 point elimination can not be achieved when applied in practical engineering. The following two examples also show that the essence is numerator.

The pole of the system is also stable on the unit circle, but if the input pole is exactly in the same position as the unit circle, it is unstable because the multiple poles produce n.

3.6 Single-sided Z-transform

Because the single-sided Z-transform only considers information that is greater than or equal to 0 parts, the same is true for two signals that are larger than the 0-part signal. One-sided Z-transform of the inverse causality signal, 0.

The single-sided Z-transform of causality signals is the same as Z-transform. The single-sided Z-transform is a time-lapse operation that is well understood, but the expression is a little trivial. The time delay negative part signal has the opportunity to top, advance, into the negative part of all were killed light

The solution of the difference equation is to consider the initial conditions, while the time-shift operation of the unilateral z-transform saves the initial conditions entirely.

The initial conditions affect the coefficients of the system's natural response and do not produce new poles. And the initial conditions have no effect on the forced response.

3.1 According to the z-transform definition Direct calculation, when the convergence domain is calculated, the absolute value of the equal-ratio factor is less than the primary.

3.2 (a) is split into U (n) and N*u (n), and NX (n) does not change the ROC. (b) The nature of the scaling is done directly in the frequency domain, and the convergence field changes. (c) Classic Easy (d) and (e) is a category, I do is E, from the Cos started to push three steps, note that a step will change the ROC. (f) E will be, F will be.

(g) n^2x (n) can be regarded as two derivative, and each time does not change the ROC (h) Easy

When seeking 0 points and extreme points, the sub-formula should be combined into a formula, otherwise the 0 points of a child may not be 0 points of the whole equation. In addition, when judging 0 points and extreme points, the semicolon is made into a positive term.

0 points and Poles: (h) 0 Point elimination to consider, when solving the root of the equation, Z is considered to be a complex number. G The answer should be wrong. F Open Although there are four cases, but can be combined into two kinds. The rest is easy.

3.3easy But it also shows how important two basic transformations are in Proakis's book.

3. 4a:z domain differential does not change ROC B: Two differential c,easy d,e easy F definition

3.5easy left, right, limited-time bilateral signal

3.6 with hint, this problem is very easy, similar to the Yu Jieyue response to get impulse response.

3.7 The direct Z-transform is multiplied, and the z-transform of the X1 is obtained in the 3.3.

3.8 A and 3.6 are repeated, B can be linear, or the time domain convolution is equal to the z-domain multiplication.

3.9 Use the nature, and then come out.

3.10 can be a normal method, you can also like the answer to design an even-numbered selection function,

3.11 The inverse transformation of z using long division divides the numerator by the denominator, which is the cause and effect, and the inverse is the inverse causality.

3.12 Using partial Fractional expansion method can be solved, tell the cause and effect signal is to do Z reverse transformation when used.

3.13 This problem is worth remembering a question, the answer is good thinking

3.14 Check the inverse of the Z-transform method, summed up here

Power Series Expansion Method: The X (z) is expanded into the form of a power series, the corresponding coefficient is the original signal. Can be used with special functions, such as e^x expansion into a power series, you can also use long division to get the expansion of power series, generally encountered on these two kinds.

Partial fractional expansion: It is generally required that the highest positive term of a molecule is less than or equal to the highest positive term of the denominator. Only solid poles in the pole, very good demolition. The conjugate single pole combines the two items together to present. Multiple poles are individually derivative.

Retention method (Inversion integration method): There are three different methods of calculation according to the ROC's three cases. The Poles are in C, leaving the numbers positive. The Poles are all outside, negative. There are inside, plus and minus two parts of the discussion. It is also important to note that the difference between the single-pole and the Bisazo-pole-left calculation method.

Http://jpkc.nwpu.edu.cn/jp2005/06/xinhaoxitong/ch8/%E7%AC%AC%E4%B8%89%E8%8A%82.htm

A, the highest term of the denominator is not lower than the highest term of the molecule, and partial fractional expansion method is used.

A, B easy

C Use the nature to get, but I used the most stupid way ... Just practice the basics ...

Deasy e does not have the Z-transform of the back triangle function, but actually does not have to do, is the trouble point.

F Using Fractional Expansion method

H according to the standard formula in the book can be seen 00 points/poles, and the remaining 0 points or Poles are a certain difference. In addition, for the trigonometric functions of special angles, it is advisable to first form the general forms.

Ieasy

J, the first form of z as a positive term, and then separate, to seek.

3.15 There are three kinds of feasible domains, double causality, double anti-causality, one causality and one anti-causality. The second obvious mistake of the answer

3.16 The resulting signal is definitely a cause-and-effect signal, so it's easy to determine when doing inverse transformations. C Inside the Cospin is ( -1) ^n.

3.17 Proof of the final value theorem I did not think of it, see the answer is barely understand, the first time to see this way of proving. Can only back down, later encountered a similar proof to understand again.

3.18

A, direct calculation by the nature of conjugate

B,c is easy to get from a, but the C book is missing a imaginary part.

D Easy Some things can be connected with 3.13

E, the left and right sides first into the same form, and then strung together.

3.19 using the differential formula of Z-domain to do directly

3.20 plus a constant g.

3.21 Real coefficients have real roots is normal, assuming that there is a complex root, the whole conjugate, you can get another complex root. In mathematics, integrals and conjugate symbols often seep into the places where they are most needed.

3.22X1 (z) *x2 (z) is the z-transform of the convolution, not the direct result of the convolution!

3.23 The power series is expanded by the Taylor formula. The factorial of 0 is 1.

3.24/25/26 Easy

3.27 according to the book. But the integral and the summation symbol interchange I remember to meet the convergence conditions.

3.28 conjugate in front of the evidence, according to the book to prove the normal product process is relatively easy to get the Wanderers theorem of the race

There is no pole in 3.29 circle C, the value is directly determined as 0

3.3030 and 50 are the same, the key one-step association is if the X1 is a root of the real coefficient polynomial, so is x1 conjugate. So the question of the requirements of the original countdown, the rest is easy

3.32 It is more convenient to solve recursive formulas with Z-transform.

3.34Z transformation is the sum of series

3.35

In the Z-transform equation There are z^7 that are returned to the time domain when the time shift is taken into consideration.

G Inside ( -1) ^n no convergence field. The answer is that the periodic signal has no convergent domain. Give another method, but I do not understand.

H, the 0 state response means that Y (n) is a causal signal, and if there is an initial condition, the effect of the initial condition needs to be considered.

3.36easy

3.37 should be unstable, because there is a pole in the unit circle, so that the unit circle will not be in the convergence region. The necessary and sufficient condition of the stable system is that the convergent domain must include the unit circle.

3.38 Easy

3.39 Easy E^jxita is to turn a corner to the plural

3.40 The minimum memory method is definitely the second method, because there will be a merge of shift registers.

3.41 the method of seeking the stable domain is also better than the book, and discusses the situation of root existence or not. In fact, the equivalent condition of the book now I do not understand.

3.42 0 Input Response The answer is better, and X (z) is considered to be 0.

3.43easy

3.44 A will simplify the form of 1/(1+Z)

b Normal Solution

3.45 the response (response) is related to the time domain.

3.46 The first one is asking for the G. The Poles are in the circle, the transformation to the time domain is the pole of the N-square, n is large when it tends to 0, the system is stable. The system's implementation is proportional to multiply, and the coefficients in front of Y (z) are replaced by 1.

3.47 cause and effect signal.

3.48 is the single-sided Z-transform of the step signal

3.49 is a simple unilateral time shift transformation, will not consider the original cause and effect signal added.

3.50 means the central symmetry, the even function is actually the symmetric value of the 2*0-n=-n and N.

3.51 C can be used with enumerations

3.52easy

3.53 feel the hint on the book is answer ....

3.54 read it in the book, you know it here.

3.55 Easy

3.56 The front is basically done with partial fractions, and here is the method of retention. Personally, the two methods have similarities in arithmetic.

3.57eaSY

3.58

The boundary of a 1,roc is determined by the radius of the pole.

2, when the n=-18, there is only one pole in C 0.5, the 0.5 can be brought in.

The answer to the first spit is really for reference only.

The anti-causality signal can be valued at 0.

H (Z) only in the 0 system is the FIR because if there is a non-0 pole, when n tends to infinity will still have a value, but the value is very small, tend to 0.

When the contour integral method is inverse Z-transform, if there is no pole in C, the direct decision is 0.

DSP Chapter III