z-Transform & Laplace Transform

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  • Laplace transform
    • a generalization of the continuous-time Fourier transform
    • converts integro-differential equations into algebraic equations
  • z-transforms
    • a generalization of the discrete-time Fourier transform
    • converts difference equations into algebraic equations

system function, transfer function: \(H(s)\)

frequency response: \(H(j\omega)\), if the ROC of \(H(s)\) includes the imaginary axis, i.e.\(s=j\omega \in \text{ROC}\)

Laplace Transform

To specify the Laplace transform of a signal, both the algebraic expression and the ROC are required. The ROC is the range of values of \(s\) for the integral of \(t\) converges

bilateral Laplace transform

image-20241001140721422

where \(s\) in the ROC and \(\mathfrak{Re}\{s\}=\sigma\)

The formal evaluation of the integral for a general \(X(s)\) requires the use of contour integration in the complex plane. However, for the class of rational transforms, the inverse Laplace transform can be determined without directly evaluating eq. (9.56) by using the technique of partial fraction expansion

ROC Property

The range of values of s for which the integral in converges is referred to as the region of convergence (which we abbreviate as ROC) of the Laplace transform

image-20241001133645466

image-20241001152307222

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i.e. no pole in RHP for stable LTI sytem

System Causality

For a causal LTI system, the impulse response is zero for \(t \lt 0\) and thus is right sided

image-20241001164006698

causality implies that the ROC is to the right of the rightmost pole, but the converse is not in general true, unless the system function is rational

System Stability

The system is stable, or equivalently, that \(h(t)\) is absolutely integrable and therefore has a Fourier transform, then the ROC must include the entire \(j\omega\)-axis

image-20241001223128791

image-20241001163547031

all of the poles have negative real parts

Unilateral Laplace transform

analyzing causal systems and, particularly, systems specified by linear constant-coefficient differential equations with nonzero initial conditions (i.e., systems that are not initially at rest)

image-20241001180202348

A particularly important difference between the properties of the unilateral and bilateral transforms is the differentiation property \(\frac{d}{dt}x(t)\)

Laplace Transform
Bilateral Laplace Transform \(sX(s)\)
Unilateral Laplace Transform \(sX(s)-x(0^-)\)

image-20241001183543726

Integration by parts for unilateral Laplace transform

image-20241001184124032

in Bilateral Laplace Transform

image-20241001231719320

In fact, the initial- and final-value theorems are basically unilateral transform properties, as they apply only to signals \(x(t)\) that are identically \(0\) for \(t \lt 0\).

\(z\)-Transform

The \(z\)-transform for discrete-time signals is the counterpart of the Laplace transform for continuous-time signals

image-20241001204402907

where \(z=re^{j\omega}\)

image-20241001204810918

The \(z\)-transform evaluated on the unit circle corresponds to the Fourier transform

image-20241001223944822

ROC Property

image-20241001222148012

system stability

The system is stable, or equivalently, that \(h[n]\) is absolutely summable and therefore has a Fourier transform, then the ROC must include the unit circle

Unilateral \(z\)-Transform

image-20241001225603923

The time shifting property is different in the unilateral case because the lower limit in the unilateral transform definition is fixed at zero, \(x[n-n_0]\)

bilateral \(z\)-transform

image-20241001232749554

unilateral \(z\)-transform

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Initial rest condition

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Initial Value Theorem & Final Value Theorem

Laplace Transform

Two valuable Laplace transform theorem

  • Initial Value Theorem, which states that it is always possible to determine the initial value of the time function \(f(t)\) from its Laplace transform \[ \lim _{s\to \infty}sF(s) = f(0^+) \]

  • Final Value Theorem allows us to compute the constant steady-state value of a time function given its Laplace transform \[ \lim _{s\to 0}sF(s) = f(\infty) \]

    If \(f(t)\) is step response, then \(f(0^+) = H(\infty)\) and \(f(\infty) = H(0)\), where \(H(s)\) is transfer function

Z-transform

Initial Value Theorem \[ f[0]=\lim_{z\to\infty}F(z) \] final value theorem \[ \lim_{n\to\infty}f[n]=\lim_{z\to1}(z-1)F(z) \]

Coert Vonk. Initial/final value proofs [https://coertvonk.com/physics/lfz-transforms/z/initial-final-value-proofs-31543]

\(s\)- and \(z\)-Domains Conversion

image-20240429220303281

image-20240429215455332

Staszewski, Robert Bogdan, and Poras T. Balsara. All-digital frequency synthesizer in deep-submicron CMOS. John Wiley & Sons, 2006.

Connection between the Laplace transform and the \(z\)-transform

image-20241002112611432

A continuous-time system with transfer function \(H(s)\) that is identical in structure to the discrete-time system \(H[z]\) except that the delays in \(H[z]\) are replaced by elements that delay continuous-time signals.

delay element Time domain Transform
\(z\)-transform \(\delta[n-1]\) \(z^{-1}\)
Laplace transform \(\delta(t-T)\) \(e^{-sT}\)
\(z\)-transform Laplace transform
\(x[n]\) \(X[z]=\sum_{n=0}^{\infty}x[n]z^{-n}\) \(\bar{x}(t)=\sum_{n=0}^{\infty}x[n]\delta(t-nT)\) \(\overline{X}(s)=\sum_{n=0}^{\infty}x[n]e^{-snT}\)
\(y[n]\) \(Y[z]=\sum_{n=0}^{\infty}y[n]z^{-n}\) \(\bar{y}(t)=\sum_{n=0}^{\infty}y[n]\delta(t-nT)\) \(\overline{Y}(s)=\sum_{n=0}^{\infty}y[n]e^{-snT}\)
\(h[n]\) \(H[z]=\sum_{n=0}^{\infty}h[n]z^{-n}\) \(\bar{h}(t)=\sum_{n=0}^{\infty}h[n]\delta(t-nT)\) \(\overline{H}(s)=\sum_{n=0}^{\infty}h[n]e^{-snT}\)
\(y[n]=x[n]*h[n]\) \(Y[z]=X[z]H[z]\) \(\bar{y}(t)=\bar{x}(t)*\bar{h}(t)\) \(\overline{Y}(s)=\overline{X}(s)\overline{H}(s)\)

Therefore, we obtain

\[\begin{align} \sum_{n=0}^{\infty}y[n]z^{-n} &=\sum_{n=0}^{\infty}x[n]z^{-n}\cdot\sum_{n=0}^{\infty}h[n]z^{-n} \\ \sum_{n=0}^{\infty}y[n]e^{-snT} &=\sum_{n=0}^{\infty}x[n]e^{-snT}\cdot \sum_{n=0}^{\infty}h[n]e^{-snT} \end{align}\]

The \(z\)-transform of a sequence can be considered to be the Laplace transform of impulses sampled train with a change of variable \(z = e^{sT}\) or \(s = \frac{1}{T}\ln z\)

Note that the transformation \(z = e^{sT}\) transforms the imaginary axis in the \(s\) plane (\(s = j\omega\)) into a unit circle in the \(z\) plane (\(z = e^{sT} = e^{j\omega T}\), or \(|z| = 1\))

Note: \(\bar{h}(t)\) is the impulse sampled version of \(h(t)\)

Note \(\bar{h}(t)\) is impulse sampled signal, whose CTFT is scaled by \(\frac{1}{T}\) of continuous signal \(h(t)\), \(\overline{H}[e^{sT}]=\overline{H}(e^{sT})\) is the approximation of continuous time system response, for example summation \(\frac{1}{1-z^{-1}}\) \[ \frac{1}{1-z^{-1}} = \frac{1}{1-e^{-sT}} \approx \frac{1}{j\omega \cdot T} \] And we know transform of integral \(u(t)\) is \(\frac{1}{s}\), as expected there is ratio \(T\)

impulse invariance

image-20241002133303153

\[ h[n] = Th_c(nT) \]

and \(T\) is chosen such that

\[ H_c(j\omega)=0, \space\space |\hat{\omega}| \ge \frac{\pi}{T} \]

When \(h[n]\) and \(h_c(t)\) are related through the above equation, i.e., the impulse response of the discrete-time system is a scaled, sampled version of \(h_c(t)\), the discrete-time system is said to be an impulse-invariant version of the continuous-time system

we have \[ H(e^{j\hat{\omega}}) = H_c\left(j\frac{\hat{\omega}}{T}\right),\space\space |\hat{\omega}| \lt \pi \]

  • \(h[n] = Th_c(nT)\) & \(T\) is small enough

    only guarantees the output equivalence only at the sampling instants, that is, \(y_c(nT) = y_r(nT)\)

  • Provided \(H_c(j\Omega)\) is bandlimited & \(T \lt 1/2f_h\)

    guarantees \(y_c(t) = y_r(t)\)

The scaling of \(T\) can alternatively be thought of as a normalization of the time domain, that is average impulse response shall be same i.e., \(h[n]\times 1 = h(nT)\times T\)

Bilinear Transformation

TODO 📅

Transfer function & sampled impulse response

continuous-time filter designs to discrete-time designs through techniques such as impulse invariance

useful functions

  • using fft

    The outputs of the DFT are samples of the DTFT

  • using freqz

    modeling as FIR filter, and the impulse response sequence of an FIR filter is the same as the sequence of filter coefficients, we can express the frequency response in terms of either the filter coefficients or the impulse response

    fft is used in freqz internally

freqz method is straightforward, which apply impulse invariance criteria. Though fft is used for signal processing mostly,

image-20241002082937903

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clear all;
clc;

%% continuous system
s = tf('s');
h = 2*pi/(2*pi+s); 	% First order lowpass filter with 3-dB frequency 1Hz
[mag, phs, wout] = bode(h);
fct = wout(:)/2/pi;
Hct_dB = 20*log10(mag(:));


fstep = 0.01;           % freq resolution
fnyqst = 1000;
Ts = 1/(2*fnyqst);
Fs = 1/Ts;              % sampling freq
Ns = ceil(Fs/fstep);    % samping points
fstep = Fs/Ns;          % update fstep
t = (0:Ns-1)*Ts;        % sampling time points

y = impulse(h, t);      % impulse resp

%% modelling as discrete system
Y = fft(y);                 % dft
Hfft = Y * Ts;              % !!! multiply Ts
Hfft_dB = 20*log10(abs(Hfft(1:Ns/2+1)));
ffft = (1:Ns/2+1)*fstep - fstep;


[Hfir, ffir] = freqz(y, 1, [], 1/Ts);   % modelling as FIR
Hfir = Hfir * Ts;           % !!! multiply Ts
Hfir_dB = 20*log10(abs(Hfir));

%% plot
semilogx(fct, Hct_dB, '-ok', ffft, Hfft_dB, 'r.-', ffir, Hfir_dB, 'b--');
legend('bode(s)', 'fft', 'FIR model')
xlabel('Freq(Hz)');
ylabel('dB');
xlim([1e-2 1e2]);
grid on;
title('frequency response of different methods');

FIR Equalization

Frequency Response

image-20220322093428287 \[ z = e^{j\omega T_s} \]

Unit impulse

filter coefficients are [-0.131, 0.595, -0.274] and sampling period is 100ps

image-20220428125454912

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%% Frequency response
w = [-0.131, 0.595, -0.274];
Ts = 100e-12;
[mag, w] = freqz(w, 1, [], 1/Ts);
subplot(2, 1, 1)
plot(w/1e9, 20*log10(abs(mag)));
xlabel('Freq(GHz)');
ylabel('dB');
grid on;
title('frequency response');

%% unit impulse response from transfer function
subplot(2, 1, 2)
z = tf('z', Ts);
h = -0.131 + 0.595*z^(-1) -0.274*z^(-2);
[y, t] = impulse(h);
stem(t*1e10, y*Ts);  % !!! y*Ts is essential
grid on;
title("unit impulse response");
xlabel('Time(\times 100ps)');
ylabel('mag');

impulse:

For discrete-time systems, the impulse response is the response to a unit area pulse of length Ts and height 1/Ts, where Ts is the sample time of the system. (This pulse approaches \(\delta(t)\) as Ts approaches zero.)

Scale output:

Multiply impulse output with sample period Ts in order to correct 1/Ts height of impulse function.

PSD transformation

If we have power spectrum or power spectrum density of both edge's absolute jitter (\(x(n)\)) , \(P_{\text{xx}}\)

Then 1UI jitter is \(x_{\text{1UI}}(n)=x(n)-x(n-1)\), and Period jitter is \(x_{\text{Period}}(n)=x(n)-x(n-2)\), which can be modeled as FIR filter, \(H(\omega) = 1-z^{-k}\), i.e. \(k=1\) for 1UI jitter and \(k=2\) Period jitter \[\begin{align} P_{\text{xx}}'(\omega) &= P_{\text{xx}}(\omega) \cdot \left| 1-z^{-k} \right|^2 \\ &= P_{\text{xx}}(\omega) \cdot \left| 1-(e^{j\omega T_s})^{-k} \right|^2 \\ &= P_{\text{xx}}(\omega) \cdot \left| 1-e^{-j\omega T_s k} \right|^2 \end{align}\]

image-20220519172239916

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clear all
close all
clc

xf_fs = 0:0.01:0.5;
k = 1;
H_1UI = 1 - exp(-1i*2*pi*xf_fs);
HH_1UI = abs(H_1UI).^2;
subplot(2, 1, 1);
plot(xf_fs, HH_1UI);
grid on;
xlabel('Freq');
ylabel('|H|^2')
title('Weight for 1UI jitter');

k = 2;
H_period = 1 - exp(-1i*2*pi*xf_fs);
HH_period = abs(H_period).^2;
subplot(2, 1, 2)
plot(xf_fs, HH_period);
grid on;
xlabel('Freq');
ylabel('|H|^2')
title('Weight for Period jitter');

image-20220709104127384 \[ x(t-\Delta T)\overset{FT}{\longrightarrow} X(s)e^{-\Delta T \cdot s} \]

Bilinear transformation

image-20241024230308374

\[\begin{align} z &= \frac{1+s\frac{T_s}{2}}{1-s\frac{T_s}{2}} \\ s &= \frac{2}{T_s}\cdot \frac{1-z^{-1}}{1+z^{-1}} \end{align}\]

where \(T_s\) is the sampling period

image-20241026230904395

reference

Alan V. Oppenheim, Alan S. Willsky, and S. Hamid Nawab. 1996. Signals & systems (2nd ed.)

Alan V Oppenheim, Ronald W. Schafer. 2010. Discrete-Time Signal Processing, 3rd edition

B.P. Lathi, Roger Green. Linear Systems and Signals (The Oxford Series in Electrical and Computer Engineering) 3rd Edition

Sam Palermo, ECEN720, Lecture 7: Equalization Introduction & TX FIR Eq

Sam Palermo, ECEN720, Lab5 –Equalization Circuits

B. Razavi, "The z-Transform for Analog Designers [The Analog Mind]," IEEE Solid-State Circuits Magazine, Volume. 12, Issue. 3, pp. 8-14, Summer 2020. [https://www.seas.ucla.edu/brweb/papers/Journals/BR_SSCM_3_2020.pdf]

Jhwan Kim, CICC 2022, ES4-4: Transmitter Design for High-speed Serial Data Communications

Mathuranathan. Digital filter design – Introduction [https://www.gaussianwaves.com/2020/02/introduction-to-digital-filter-design/]

Daniel Boschen, "Fast Track to Designing FIR Filters with Python" [https://events.gnuradio.org/event/24/contributions/598/attachments/186/485/Boschen%20FIR%20Filter%20Presentation.pdf]

Daniel Boschen, "Quick Start on Control Loops with Python" [https://events.gnuradio.org/event/24/contributions/599/attachments/187/480/Boschen%20Control%20Presentation.pdf]