Signal and System insight
dBc
TODO π
dBFS
TODO π
Nyquist rate & Nyquist frequency
Nyquist rate
The Nyquist rate is the minimum sample rate required to accurately measure a signal's highest frequency. It's equal to twice the highest frequency of the signal
Nyquist frequency
The Nyquist frequency is the highest frequency that can be represented without aliasing in a discrete signal. It's equal to half the sampling frequency
Oversampling Ratio (OSR) is defined as the ratio of the Nyquist frequency \(f_s/2\) to the signal bandwidth \(B\) given by \(\text{OSR}=f_s/2B\)
Summation & Integration
| impulse response | Transform | ROC | |
|---|---|---|---|
| Summation | \(u(t)\) | \(\frac{1}{s}\) | \(\mathfrak{Re}\{s\}\gt 0\) |
| Integration | \(u[n]\) | \(\frac{1}{1-z^{-1}}\) | \(|z| \gt 1\) |
both are NOT stable
sinc filter
where \(W\) is sampling frequency in Hz
Zero-order hold (ZOH)
\[
h_{ZOH}(t) = \text{rect}(\frac{t}{T} - \frac{1}{2}) = \left\{
\begin{array}{cl}
1 & : \ 0 \leq t \lt T \\
0 & : \ \text{otherwise}
\end{array} \right.
\] The effective frequency response is the continuous Fourier
transform of the impulse response \[
H_{ZOH}(f) = \mathcal{F}\{h_{ZOH}(t)\} = T\frac{1-e^{j2\pi fT}}{j2\pi
fT}=Te^{-j\pi fT}\text{sinc}(fT)
\] where \(\text{sinc}(x)\) is
the normalized sinc function \(\frac{\sin(\pi
x)}{\pi x}\)
The Laplace transform transfer function of the ZOH is found by substituting \(s=j2\pi f\) \[ H_{ZOH}(s) = \mathcal{L}\{h_{ZOH}(t)\}=\frac{1-e^{-sT}}{s} \]
frequency convention
- radian frequency \(\omega_0\) in rad/s
- cyclic frequency \(f_0\) in Hz
Energy signals vs Power signal
Topic 5 Energy & Power Signals, Correlation & Spectral Density [https://www.robots.ox.ac.uk/~dwm/Courses/2TF_2021/N5.pdf]
modulation & demodulation
Hossein Hashemi, RF Circuits, [https://youtu.be/0f3yZMvD2Jg?si=2c1Q4y6WJq8Jj8oN]
Coherent Sampling
To avoid spectral leakage completely, the method of coherent sampling is recommended. Coherent sampling requires that the input- and clock-frequency generators are phase locked, and that the input frequency be chosen based on the following relationship: \[ \frac{f_{\text{in}}}{f_{\text{s}}}=\frac{M_C}{N_R} \]
where:
- \(f_{\text{in}}\) = the desired input frequency
- \(f_s\) = the clock frequency of the data converter under test
- \(M_C\) = the number of cycles in the data window (to make all samples unique, choose odd or prime numbers)
- \(N_R\) = the data record length (for an 8192-point FFT, the data record is 8192s long)
\[\begin{align} f_{\text{in}} &=\frac{f_s}{N_R}\cdot M_C \\ &= f_{\text{res}}\cdot M_C \end{align}\]
irreducible ratio
An irreducible ratio ensures identical code sequences not to be repeated multiple times. Unnecessary repetition of the same code is not desirable as it increases ADC test time.
Given that \(\frac{M_C}{N_R}\) is irreducible, and \(N_R\) is a power of 2, an odd number for \(M_C\) will always produce an irreducible ratio
Assuming there is a common factor \(k\) between \(M_C\) and \(N_R\), i.e. \(\frac{M_C}{N_R}=\frac{k M_C'}{k N_R'}\)
The samples (\(n\in[1, N_R]\))
\[\begin{align} y[n] &= \sin\left( \omega_{\text{in}} \cdot t_n \right) \\ &= \sin\left( \omega_{\text{in}} \cdot n\frac{1}{f_s} \right) \\ & = \sin\left( \omega_{\text{in}} \cdot n\frac{1}{f_{\text{in}}}\frac{M_C}{N_R} \right) \\ & = \sin\left( 2\pi n\frac{M_C}{N_R} \right) \end{align}\]
Then
\[\begin{align} y[n+N_R'] &= \sin\left( 2\pi (n+N_R')\frac{M_C}{N_R} \right) \\ & = \sin\left( 2\pi n \frac{M_C}{N_R} + 2\pi N_R'\frac{M_C}{N_R}\right) \\ & = \sin\left( 2\pi n \frac{M_C}{N_R} + 2\pi N_R'\frac{kM_C'}{kN_R'} \right) \\ & = \sin\left( 2\pi n \frac{M_C}{N_R} + 2\pi M_C' \right) \\ & = \sin\left( 2\pi n \frac{M_C}{N_R}\right) \end{align}\]
So, the samples is repeated \(y[n] = y[n+N_R']\). Usually, no additional information is gained by repeating with the same sampling points.
Example \[ N \cdot \frac{1}{F_s} = M \cdot \frac{1}{F_{in}} \]
where \(F_s\) is sample frequency, \(F_{in}\) input signal frequency.
And \(N\) often is 256, 512; M is 3, 5, 7, 11.
channel loss
- skin effect loss
- dielectric loss
phase delay & group delay
- Phase delay directly measures the device or system time delay of individual sinusoidal frequency components in the steady-state conditions.
- In the ideal case the envelope delay is equal to the phase delay
- envelope delay is a more sensitive measure of aberrations than phase delay
phase delay
If the phase delay peaks (exceeds the low-frequency value) you can expect to see high-frequency components late in the step response. This causes ringing.
group delay
steady-state at this frequency is a polarity flip; a 180 degrees phase shift; which is a transfer function of H(s)=-1. \[ H(s) = e^{j\pi} \] That is \(\phi(\omega) = \pi\) \[ \tau_p = \frac{\pi}{\omega} \] and \[ \tau_g = \frac{\partial \pi}{\partial \omega}=0 \]
Hollister, Allen L. Wideband Amplifier Design. Raleigh, NC: SciTech Pub., 2007.
Pupalaikis, Peter. (2006). Group Delay and its Impact on Serial Data Transmission and Testing. [https://cdn.teledynelecroy.com/files/whitepapers/group_delay-designcon2006.pdf]
[Pupalaikis et al., βEye Patterns in Scopesβ, DesignCon, Santa Clara CA, 2005https://cdn.teledynelecroy.com/files/whitepapers/eye_patterns_in_scopes-designcon_2005.pdf]
StariΔ, P. & Margan, E.. (2006). Wideband Amplifiers. 10.1007/978-0-387-28341-8.
Alan V. Oppenheim, Alan S. Willsky, and S. Hamid Nawab. 1996. Signals & systems (2nd ed.). Prentice-Hall, Inc., USA.
Phase delay vs group delay: Common misconceptions. [https://audiosciencereview.com/forum/index.php?threads/phase-delay-vs-group-delay-common-misconceptions.39591/]
Feedback Rearrange
The closed loop transfer function of \(Y/X\) and \(Y_1/X_1\) are almost same, except sign
\[\begin{align} \frac{Y}{X} &= +\frac{H_1(s)H_2(s)}{1+H_1(s)H_2(s)} \\ \frac{Y_1}{X_1} &= -\frac{H_1(s)H_2(s)}{1+H_1(s)H_2(s)} \end{align}\]
define \(-Y_1=Y_n\), then \[
\frac{Y_n}{X_1} = \frac{H_1(s)H_2(s)}{1+H_1(s)H_2(s)}
\] 
Saurabh Saxena, IIT Madras. CICC2022 Clocking for Serial Links - Frequency and Jitter Requirements, Phase-Locked Loops, Clock and Data Recovery
Convolution of probability distributions
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions.
Thermal noise
Thermal noise in an ideal resistor is approximately white, meaning that its power spectral density is nearly constant throughout the frequency spectrum.
When limited to a finite bandwidth and viewed in the time domain, thermal noise has a nearly Gaussian amplitude distribution
Barkhausen criteria
Barkhausen criteria are necessary but not sufficient conditions for sustainable oscillations
it simply "latches up" rather than oscillates
NRZ Bandwidth
Maxim Integrated,NRZ Bandwidth - HF Cutoff vs. SNR [https://pdfserv.maximintegrated.com/en/an/AN870.pdf]
\(0.35/T_r\)
\(0.5/T_r\)
TODO π
System Type
Control of Steady-State Error to Polynomial Inputs: System Type
control systems are assigned a type number according to the maximum degree of the input polynominal for which the steady-state error is a finite constant. i.e.
- Type 0: Finite error to a step (position error)
- Type 1: Finite error to a ramp (velocity error)
- Type 2: Finite error to a parabola (acceleration error)
The open-loop transfer function can be expressed as \[ T(s) = \frac{K_n(s)}{s^n} \]
where we collect all the terms except the pole (\(s\)) at eh origin into \(K_n(s)\),
The polynomial inputs, \(r(t)=\frac{t^k}{k!} u(t)\), whose transform is \[ R(s) = \frac{1}{s^{k+1}} \]
Then the equation for the error is simply \[ E(s) = \frac{1}{1+T(s)}R(s) \]
Application of the Final Value Theorem to the error formula gives the result
\[\begin{align} \lim _{t\to \infty} e(t) &= e_{ss} = \lim _{s\to 0} sE(s) \\ &= \lim _{s\to 0} s\frac{1}{1+\frac{K_n(s)}{s^n}}\frac{1}{s^{k+1}} \\ &= \lim _{s\to 0} \frac{s^n}{s^n + K_n}\frac{1}{s^k} \end{align}\]
- if \(n > k\), \(e=0\)
- if \(n < k\), \(e\to \infty\)
- if \(n=k\)
- \(e_{ss} = \frac{1}{1+K_n}\) if \(n=k=0\)
- \(e_{ss} = \frac{1}{K_n}\) if \(n=k \neq 0\)β
where we define \(K_n(0) = K_n\)
Nyquist's Stability Criterion
TODO π
Spectral content of NRZ
Lecture 26 Autocorrelation Functions of Random Binary Processes [https://bpb-us-w2.wpmucdn.com/sites.gatech.edu/dist/a/578/files/2003/12/ECE3075A-26.pdf]
Lecture 32 Correlation Functions & Power Density Spectrum, Cross-spectral Density [https://bpb-us-w2.wpmucdn.com/sites.gatech.edu/dist/a/578/files/2003/12/ECE3075A-32.pdf]
sinusoidal steady-state and frequency response
Due to KCL and \(u(t)=e^{j\omega t}\) and \(y(t)=H(j\omega)e^{j\omega t}\), we have ODE:
\[\begin{align} \frac{u(t) - y(t)}{R} = C \frac{dy(t)}{dt} \\ e^{j\omega t} - H(j\omega) e^{j\omega t} = H(j\omega)\cdot j\omega e^{j\omega t} \\ \end{align}\]
\(H(j\omega)\) is obtained as below \[ H(j\omega) = \frac{1}{1+j\omega} \]
Different Variants of the PSD Definition
In the practice of engineering, it has become customary to use slightly different variants of the PSD definition, depending on the particular application or research field.
Two-Sided PSD, \(S_x(f)\)
this is a synonym of the PSD defined as the Fourier Transform of the autocorrelation.
One-Sided PSD, \(S'_x(f)\)
this is a variant derived from the two-sided PSD by considering only the positive frequency semi-axis.
To conserve the total power, the value of the one-sided PSD is twice that of the two-sided PSD \[ S'_x(f) = \left\{ \begin{array}{cl} 0 & : \ f \geq 0 \\ S_x(f) & : \ f = 0 \\ 2S_x(f) & : \ f \gt 0 \end{array} \right. \]
Note that the one-sided PSD definition makes sense only if the two-sided is an even function of \(f\)
If \(S'_x(f)\) is even symmetrical around a positive frequency \(f_0\), then two additional definitions can be adopted:
Single-Sideband PSD, \(S_{SSB,x}(f)\)
This is obtained from \(S'_x(f)\) by moving the origin of the frequency axis to \(f_0\) \[ S_{SSB,x}(f) =S'_x(f+f_0) \] This concept is particularly useful for describing phase or amplitude modulation schemes in wireless communications, where \(f_0\) is the carrier frequency.
Note that there is no difference in the values of the one-sided versus the SSB PSD; it is just a pure translation on the frequency axis.
Double-Sideband PSD, \(S_{DSB,x}(f)\)
this is a variant of the SSB PSD obtained by considering only the positive frequency semi-axis.
As in the case of the one-sided PSD, to conserve total power, the value of the DSB PSD is twice that of the SSB \[ S_{DSB,x}(f) = \left\{ \begin{array}{cl} 0 & : \ f \geq 0 \\ S_{SSB,x}(f) & : \ f = 0 \\ 2S_{SSB,x}(f) & : \ f \gt 0 \end{array} \right. \]
Note that the DSB definition makes sense only if the SSB PSD is even symmetrical around zero
Poles and Zeros of transfer function
poles
\[ H(s) = \frac{1}{1+s/\omega_0} \]
magnitude and phase at \(\omega_0\) and \(-\omega_0\) \[\begin{align} H(j\omega_0) &= \frac{1}{1+j} = \frac{1}{\sqrt{2}}e^{-j\pi/4} \\ H(-j\omega_0) &= \frac{1}{1-j} = \frac{1}{\sqrt{2}}e^{j\pi/4} \end{align}\]
system response \(y(t)\) of input \(\cos(\omega_0 t)\), note \(\cos(\omega_0t) = \frac{1}{2}(e^{j\omega_0 t} + e^{-j\omega_0 t})\) \[\begin{align} y(t) &= H(j\omega_0)\cdot \frac{1}{2}e^{j\omega_0 t} + H(-j\omega_0)\cdot \frac{1}{2}e^{-j\omega_0 t} \\ &= \frac{1}{\sqrt{2}}\cos(\omega_0t-\pi/4) \end{align}\]
\(\cos(\omega_0 t)\), with frequency same with pole DON'T have infinite response
That is, pole indicate decrease trending
zeros
similar with poles, \(\cos(\omega_0 t)\), with frequency same with zero DON'T have zero response
\[ H(s) = 1+s/\omega_0 \]
magnitude and phase at \(\omega_0\) and \(-\omega_0\) \[\begin{align} H(j\omega_0) &= 1+j = \sqrt{2}e^{j\pi/4} \\ H(-j\omega_0) &= 1-j = \sqrt{2}e^{-j\pi/4} \end{align}\]
system response \(y(t)\) of input \(\cos(\omega_0 t)\), note \(\cos(\omega_0t) = \frac{1}{2}(e^{j\omega_0 t} + e^{-j\omega_0 t})\) \[\begin{align} y(t) &= H(j\omega_0)\cdot \frac{1}{2}e^{j\omega_0 t} + H(-j\omega_0)\cdot \frac{1}{2}e^{-j\omega_0 t} \\ &= \sqrt{2}\cos(\omega_0t+\pi/4) \end{align}\]
baud rate
symbol rate, modulation rate or baud rate is the number of symbol changes per unit of time.
- Bit rate refers to the number of bits transmitted between two devices per unit of time
- The baud or symbol rate refers to the number of symbols that can be sent in the same amount of time
reference
Stephen P. Boyd. EE102 Lecture 10 Sinusoidal steady-state and frequency response [https://web.stanford.edu/~boyd/ee102/freq.pdf]
Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini. 2018. Feedback Control of Dynamic Systems (8th Edition) (8th. ed.). Pearson.
Inter-Symbol Interference (or Leaky Bits) [http://blog.teledynelecroy.com/2018/06/inter-symbol-interference-or-leaky-bits.html]
[AN001] Designing from zero an IIR filter in Verilog using biquad structure and bilinear discretization. URL:[https://www.controlpaths.com/articles/an001_designing_iir_biquad_filter_bilinear/]
Frequency warping using the bilinear transform. URL:[https://www.controlpaths.com/2022/05/09/frequency-warping-using-the-bilinear-transform/]
Digital control loops. Theoretical approach. URL:[https://www.controlpaths.com/2022/02/28/digital-control-loops-theoretical-approach/]
Simulation of DSP algorithms in Verilog. URL:[https://www.controlpaths.com/2023/05/20/simulation-of-dsp-algorithms-in-verilog/]
Implementing a digital biquad filter in Verilog. URL:[https://www.controlpaths.com/2021/04/19/implementing-a-digital-biquad-filter-in-verilog/]
Implementing a FIR filter using folding. URL:[https://www.controlpaths.com/2021/05/17/implementing-a-fir-filter-using-folding/]
Oppenheim, Alan V. and Cram. βDiscrete-time signal processing : Alan V. Oppenheim, 3rd edition.β (2011).
Extras: PID Compensator with Bilinear Approximation URL:[https://ctms.engin.umich.edu/CTMS/index.php?aux=Extras_PIDbilin]