Signal and System insight

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convolution, cross-correlation & autocorrelation

[https://upload.wikimedia.org/wikipedia/commons/2/21/Comparison_convolution_correlation.svg]

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Hilbert Transform

Mathuranathan. Understanding Analytic Signal and Hilbert Transform [https://www.gaussianwaves.com/2017/04/analytic-signal-hilbert-transform-and-fft/]

Dan Boschen. What information does the Hilbert transform give? [https://dsp.stackexchange.com/a/88292/59253]

Frank R. Kschischang. The Hilbert Transform [https://www.comm.utoronto.ca/frank/notes/hilbert.pdf]

Derek Rowell, 2.161 Signal Processing: Continuous and Discrete Fall 2008: Determining a System’s Causality from its Frequency Response [https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/142cd928b3b3959721198872ab97b647_causality.pdf]

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In frequency domain \[ \boxed{\mathcal{H}\{X(\omega)\} = \frac{1}{\pi\omega} * X(\omega)} \qquad \boxed{\mathcal{H}\{x(f)\} = \frac{1}{\pi f} * x(f)} \]

image-20260619081200833

Pulse Amplitude Modulation (PAM)

Qasim Chaudhari. Pulse Amplitude Modulation (PAM) [https://wirelesspi.com/pulse-amplitude-modulation-pam/]

TODO 📅

pulse averaging

The average value of output pulse are same because of same DC gain (0dB)

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Pulse shape vs Frequency Magnitude

TODO 📅

Incrementally linear system

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fft vs. ifft

Jason Sachs, Ten Little Algorithms, Part 2: The Single-Pole Low-Pass Filter [https://www.embeddedrelated.com/showarticle/779.php]

image-20250907005625071

\[ \mathcal{ifft} = \frac{\mathcal{fft}}{N} \] image-20250907005201992

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import matplotlib.pyplot as plt
import numpy as np
np.random.seed(123456789)   # repeatable results

f0 = 4
t = np.arange(0,1.0,1.0/65536)
mysignal = (np.mod(f0*t,1) < 0.5)*2.0-1
mynoise = 1.0*np.random.randn(*mysignal.shape)

plt.figure(figsize=(8,6))
plt.plot(t, mysignal+mynoise, 'gray',
         t, mysignal,         'black');


def spectrum_fftovN(x):
    return np.abs(np.fft.fft(x))/len(x)

def spectrum_ifft(x):
    return np.abs(np.fft.ifft(x))


mysignal_spectrum_fftovN = spectrum_fftovN(mysignal)
mynoise_spectrum_fftovN  = spectrum_fftovN(mynoise)
mysignal_spectrum_ifft= spectrum_ifft(mysignal)
mynoise_spectrum_ifft = spectrum_ifft(mynoise)

N1 = 500
f = np.arange(N1)
plt.figure(figsize=(16,12))
plt.subplot(2,1,1)
plt.plot(f,mysignal_spectrum_fftovN[:N1], 'b-',
         f,mysignal_spectrum_ifft[:N1], 'r--', linewidth=3)
plt.legend(('fftovN','ifft'), fontsize=16, loc='upper right')
plt.title('signal', fontsize=24); plt.xlim(0,N1); plt.xlabel('frequency'); plt.ylabel('amplitude')

plt.subplot(2,1,2)
plt.plot(f,mynoise_spectrum_fftovN[:N1], 'b-',
         f,mynoise_spectrum_ifft[:N1], 'r--', linewidth=3)
plt.legend(('fftovN','ifft'), fontsize=16, loc='upper right')
plt.title('noise', fontsize=24); plt.xlim(0,N1); plt.xlabel('frequency'); plt.ylabel('amplitude')

plt.show()

Pulse Code Modulation (PCM)

John M Pauly. Lecture 13: Pulse Code Modulation [https://web.stanford.edu/class/ee179/lectures/notes13.pdf]

Pulse Code Modulation (PCM) is a method for digitally representing analog signals by sampling their amplitude at regular intervals and then encoding these samples into binary numbers

image-20250820222558595

Energy/bit (pJ/b)

1mW/Gbps = 1pJ/bit

Joules are a unit of work or energy. Watts are a unit of power which is the rate at which energy is generated or consumed.

modulation depth

The modulation index (or modulation depth) of a modulation scheme describes by how much the modulated variable of the carrier signal varies around its unmodulated level

TODO 📅

Image frequency

Antonio Liscidini, ESSCIRC 2019 Tutorials: Ultra Low Power Receivers [https://youtu.be/OJRB8g4vUZw]

TODO 📅

RMS for non-sinusoidal periodic function

image-20241220214234185

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

https://upload.wikimedia.org/wikipedia/commons/d/d8/Nyquist_frequency_%26_rate.svg

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

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]

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modulation & demodulation

Hossein Hashemi, RF Circuits, [https://youtu.be/0f3yZMvD2Jg]

image-20240826221237312

image-20240826221251379

Convolution of probability distributions

The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions.

image-20240804104528903

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

image-20240804102454281

sinusoidal steady-state and frequency response

image-20231104104933781

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Due to KCL and \(u(t)=e^{j\omega t}\) and \(y(t)=H(j\omega)e^{j\omega t}\), we have ODE:

\[\begin{align} \frac{u(t) - y(t)}{R} = C \frac{\mathrm{d}y(t)}{\mathrm{d}t} \\ e^{j\omega t} - H(j\omega) e^{j\omega t} = H(j\omega)\cdot j\omega e^{j\omega t} \\ \end{align}\]

\(H(j\omega)\) is obtained as below \[ H(j\omega) = \frac{1}{1+j\omega} \]

image-20231104135855739

PSD Definition Variants

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 \[ S'_x(f) = \left\{ \begin{array}{cl} 0 & : \ f \geq 0 \\ S_x(f) & : \ f = 0 \\ 2S_x(f) & : \ f \gt 0 \end{array} \right. \]

image-20230603185546658

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 \[ S_{DSB,x}(f) = \left\{ \begin{array}{cl} 0 & : \ f \geq 0 \\ S_{SSB,x}(f) & : \ f = 0 \\ 2S_{SSB,x}(f) & : \ f \gt 0 \end{array} \right. \]

image-20230603222054506

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. “Discrete-time signal processing : Alan V. Oppenheim, 3rd edition.” (2011).

Extras: PID Compensator with Bilinear Approximation URL:[https://ctms.engin.umich.edu/CTMS/index.php?aux=Extras_PIDbilin]