Discrete Time System in transformed domain

Some actual signals do not have Fourier transformation. Just like the introduction of Laplace transformation, adding an attenuation factor will satisfy the conditions. The ing from Laplace to ztransformation can be understood as continuous to discrete.

Ztransformation is an infinite series, and the infinite series has a convergence domain problem. It can be understood that the convergence domain is the region where Fourier transformation exists.

The partial fraction method of the inverse transformation of the ztransform can be calculated using functions: [R, P, C] = residuez (B,)

Where B and A are the coefficients vectors of the polynomials arranged by the ascending power sequence of the Z-1 and the grave;

R is the left number vector of each root, and P is the pole vector. C.

You can also use H = impz (B, A, N ). We have previously introduced how to obtain h (n) from the denominator of a known polynomial. That is to say, we can perform inverse transformation.

As for solving the difference equation, we have previously introduced filter (B, A, X, xic). xic is the input sequence of the initial condition.

Calculation of the initial condition: xic = filtic (B, A, Y, X)

B and A are arrays of denominator coefficients. Y and X are the initial condition arrays, y = [Y (-1), y (-2),...], X = [x (-1), x (-2)...].

Next we will talk about the spectral analysis on the zplane.

I have learned dtft geometric method before. It can be found that if the pole is very close to the unit circle, a large harmonic peak will appear near the pole, and the denominator will be quickly reduced.

Due to stability requirements, the pole must be within the unit circle, which is interpreted as a negative phase shift. When the zero point is also within the Unit Circle, the negative phase shift of the system is the least (zero point can produce a positive phase shift offset), called the minimum phase system.

Spectrum Analysis on non-unit circumference.

For example, in speech signal processing, we often need to know the frequency of the pole. If the unit circle in the pole is far away, the spectrum on the unit circle is smooth.

If the trajectory of the sampling point is carried out along an arc or circumference close to these poles, the sampling result will show a significant spike at the frequency corresponding to the pole.

The pulse response of an ideal filter is a SA function. For the sake of cause and effect, only n> = 0 can be intercepted. Considering the linear phase requirements, the intercepted sequence must be symmetric.

In order to make it closer to the ideal situation, we should increase the latency as much as possible and increase the truncation length (order ).

The shorter the truncation sequence, the larger the difference between the amplitude-frequency characteristics and ideal conditions.

If the intercepted sequence is symmetric, the phase frequency is linear. If asymmetry occurs, the phase-frequency characteristics are non-linear.

Use a zero pole analysis filter.

The rule is: the closer the frequency is to zero, the smaller the amplitude. The closer the frequency to the pole, the larger the amplitude.

From z = eiw, Z =-1 to the lowest frequency. Therefore, zero point Z =-1 can obtain a higher low frequency range.

After Z =-1, for the first-order low-pass filter, the relationship between the channel width and pole a is approximately Wp = 1-A. Note that WP is a digital frequency.

Second-order is more flexible. For filtering or ripple, you can directly set the zero point to ejw0 on the unit circle of this angle frequency.

Similarly, the comb filter distributes zero points evenly on the unit circle. The pole position is very close to the zero point, which can narrow the ripple feature.

However, the phase-frequency characteristics of the trap filter are not good. It is generally necessary to enable the cascade full-pass filter for correction.

Discrete Time System in transformed domain