Complex-trace analysis

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Problem 9.26a

Given the wavelet $ [10,\;8,\;0,\;-9,\;-11,\;-6,\;0,\;7,\;12,\;5,\;0,\;0] $, calculate the quadrature function, $ g_{\bot }(t) $.

Background

A wave with continuously varying amplitude is of the form


$ {\begin{aligned}g(t)=A(t){\rm {\;cos\;}}\omega t,\end{aligned}} $ (9.26a)

where we assume that the rate of change of $ A(t) $ is small compared with $ \omega $ and the sampling frequency $ 1/\Delta $. If we take $ A(t) $ fixed, the Hilbert transform (see Sheriff and Geldart, 1995, section 15.2.13) of $ g(t) $, written $ g_{\bot }(t) $, is


$ {\begin{aligned}g_{\bot }(t)=-A(t)\sin \omega t.\end{aligned}} $ (9.26b)

We note that $ g(t) $ and $ g_{\bot }(t) $ differ in phase by $ -90^{\circ } $. We can combine the signals in equations (9.26a,b) to obtain the complex function $ h(t) $:


$ {\begin{aligned}h(t)=g(t)+{\hbox{j}}g_{\bot }(t)=A(t)e^{-{\hbox{j}}\omega t}\end{aligned}} $ (9.26c)

(see Sheriff and Geldart, 1995, problem 15.12a for Euler’s formulas and Sheriff and Geldart, 1995, section 15.1.5 for a discussion of complex functions). The quantities $ h(t) $ and $ g_{\bot }(t) $ are known as the complex trace and the quadrature trace, respectively. $ A(t) $ and its mirror image constitute the envelope of both $ g(t) $ and $ g_{\bot }(t) $.


$ {\begin{aligned}A(t)=[g^{2}(t)+g_{\bot }^{2}(t)]^{1/2}\end{aligned}} $ (9.26d)

We think of the complex trace as being traced by the tip of a vector of length $ A(t) $ rotating in the complex plane as it moves perpendicular to the complex plane in the time direction (Figure 9.26a). The projection of the helical path generated by the tip of this vector onto the real plane is $ g(t) $ and the projection onto the imaginary plane is $ g_{\bot } $. The angle that this vector makes with the real plane is the instantaneous phase $ \gamma (t) $ and the rotational speed is the instantaneous frequency $ f_{i}=d\gamma /dt $.


$ {\begin{aligned}\gamma (t)&=\tan ^{-1}\left({\frac {g_{\bot }(t)}{g(t)}}\right).\end{aligned}} $ (9.26e)


$ {\begin{aligned}f(t)&=d\gamma (t)/dt={\frac {g(t){\frac {dg_{\bot }(t)}{dt)}}-g_{\bot }(t){\frac {dg(t)}{dt)}}}{g^{2}(t)+g_{\bot }^{2}(t)}}\approx {\frac {\Delta \gamma }{\Delta t}}={\frac {\gamma _{i+1}-\gamma _{i-1}}{2\Delta }}.\end{aligned}} $ (9.26f)
Figure 9.26a.  Graphs of $ g_{t} $ (dashed line), $ g_{\bot t} $ (solid line), and $ \pm A $ (dotted lines).

Because $ tan\theta ={\rm {\;tan\;}}(\theta +n\pi ) $, $ n $ integral, we generally add (or subtract) multiples of $ \pi $ to $ \gamma (t) $ to make it monotonically increasing (or decreasing). The derivative of the arc tangent in equation (9.26f) usually permits more stable calculation than taking the derivative of $ \gamma (t) $ itself. However, the finite-difference expression is often sufficiently accurate.

To get $ g_{\bot }(t) $ we find the Hilbert transform of $ g(t) $ [see Sheriff and Geldart, 1995, equation (15.176)]:

$ {\begin{aligned}g_{\bot }(t)=g(t)*{\frac {-1}{\pi t}}.\end{aligned}} $

For a digital function $ g_{t} $, this becomes [see Sheriff and Geldart, 1995, equation (9.107)]


$ {\begin{aligned}g_{\bot t}=g_{t}*q_{t}=\mathop {\sum } {_{n=-\infty }^{+\infty }}g_{t-n}(e^{{\hbox{j}}n\pi }-1)/\pi n,\end{aligned}} $ (9.26g)

where $ q_{t} $ is the quadrature filter. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (e^{\hbox {j}n\pi} -1)=0 or –2 according as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n is odd or even, the equation reduces to


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} g_{\bot t} =\left(\frac{-2}{\pi}\right)=\mathop{\sum}{_{n=-\infty}^{+\infty}} g_{t-n} /n,\quad \mathrm{n\ odd}. \end{align} (9.26h)

Solution

We note that 11 samples constitute 5/4 cycles so, assuming 2-ms sampling, the period is 20(4/5) = 16 ms or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f = 62 Hz = dominant frequency.

While Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g_{t} is causal with elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g_{0} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g_{9} , equation (9.26h) shows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g_{\bot t} is not causal. We use equation (9.26h) to calculate $ g_{\bot t} $ for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t=0 to 9. Thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} g_{\bot 0}&=(-2/\pi)[g_{1} /(-1)+g_{3} /(-3)+g_{5} /(-5)+g_{7} /(-7)+g_{9} (-9)]\\ &=0.64(8/1-9/3-6/5+7/7+5/9)=0.64\times 5.36=3.43;\\ g_{\bot 1}&=0.64(g_{0} /1+g_{2} /(-1)+g_{4} /(-3)+g_{6} /(-5)+g_{8} /(-7)=0.64\\ &=0.64[(0+0-11/(-3)+0+12/(-7)]=0.64\times 11.95=7.65. \end{align}

Continuing the calculation as far as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g_{\bot 9} , we obtain for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g_{\bot t}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} g_{\bot t}=\left[\mathop{-3.4}\limits^{\downarrow},\; 7.2,\; 10.8,\; 7.6,\; -2.3\; ,\; -8.3\; ,\; -10.3\; ,\; -9.1\; ,\; -0.4,\; 7.0\right]. \end{align}

Figure 9.26a shows Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g_{t} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g_{\bot t} .

Problem 9.26b

What are values of the complex trace Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h ($ t $) and the amplitude of the envelope Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A(t) .

Solution

Using equations (9.26c,d), we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} h_{0} =10-3.4j\qquad A_{0} =10.6\\ h_{1} =8+7.2j\qquad A_{1} =10.8\\ h_{2} =10.8j\qquad A_{2} =10.8\\ h_{3} =-9+7.6j\qquad A_{3} =11.8\\ h_{4} =-11-2.3j\qquad A_{4} =11.3\\ h_{5} =-6-8.3j\qquad A_{5} =10.2\\ h_{6} =-10.3j\qquad A_{6} =10.3\\ h_{7} =7-9.1j\qquad A_{7} =11.4\\ h_{8} =12-0.4j\qquad A_{8} =12.0\\ h_{9} =5+7.0j\qquad A_{9} =8.6 \end{align}

Figure 9.26b.  Graph of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{t} .

Figure 9.26b is a graph of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{t} ;h_{0} is in the southeast quadrant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{2} in the northeast, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{3} in the northwest, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{4} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{5} , and $ h_{6} $ in the southwest, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{7} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{8} in the southeast, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{9} in the northeast. Thus the phase makes a little over one revolution, just as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{t} does.

The amplitude Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A_{t} is shown dotted in figure 9.26a.

Problem 9.26c

Calculate the instantaneous phase $ \gamma (t) $ and the instantaneous frequency Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(t) (adding to the phase multiples of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \pi as necessary to obtain a monotonically increasing function. Assume Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta =2 ms.

Solution

We use equations (9.26e,f) to obtain values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \gamma_{t} and $ f_{t} $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i=0 to 9. To calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{t} using the derivative operator (problem 9.31) in equation (9.26f) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i=0 to 9, we need Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \gamma _{-1} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \gamma _{10} . Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g_{1} =0=g_{10} , equation (9.26e) shows that both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \gamma _{-1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \gamma _{10} are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \pm 90^{\circ} , depending on the signs of $ g_{\bot (-1)} $ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g_{\bot 10} ; to determine these signs we use equations (9.26h) to write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} g_{\bot (-1)}&=\frac{-2}{\pi} \mathop{\sum}\nolimits_{-\infty}^{+\infty} \frac{g_{-1-n}}{n} \\ &=\frac{2}{\pi} \mathop{\sum}\limits_{}^{} \frac{g_{-1-n}}{-n} ,\; n=-1,\; -3,\; -5,\; -7,\; -9\\ &=0.64(g_{0} /1+g_{2} /3+g_{4} /5+g_{6} /7+g_{8} /9)\\ &=0.64(10/1-11/5+12/9)= \mathrm{positive},\\ g_{\bot 10} &=-0.64(g_{9} /1+g_{7} /3+g_{5} /5+g_{3} /7+g_{1} /9)\\ &=-0.64(5/1+7/3-6/5-9/7+8/9)= \mathrm{negative}. \end{align}

Thus, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm \; tan\;}\gamma _{-1} =+\infty , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \gamma _{-1} =90^{\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm \; tan\;}\gamma _{10} =-\infty , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \gamma _{10} =-90^{\circ} .

We now calculate the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \gamma _{i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{i} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i=0 to 9.

$ {\begin{aligned}\gamma _{0}&={\tan }^{-1}\left({\frac {g_{\bot 0}}{g_{0}}}\right)={\tan }^{-1}(-3.4/10)&f_{0}=\{\gamma _{1}-\gamma _{-1})/2\times 0.002\times 360^{*}\\&=-19^{\circ };+180^{\circ }=161^{\circ }&=132/1.44=92\mathrm {Hz} \\\gamma _{1}&={\tan }^{-1}[(7.2/8]&f_{1}=109/1.44=76\ \mathrm {Hz} \\&=42^{\circ }+180^{\circ }=222^{\circ }&\\\gamma _{2}&=90^{\circ }+180^{\circ }=270^{\circ }&f_{2}=98/1.44=68\mathrm {Hz} \\\gamma _{3}&={\tan }^{-1}[7.6/(-9)]&f_{3}=102/1.44=71\mathrm {Hz} \\&=-40^{\circ }+360^{\circ }=320^{\circ }&\\\gamma _{4}&={\tan }^{-1}[-2.3/(-11)]&f_{4}=94/1.44=65\mathrm {Hz} \\&=12^{\circ }+360^{\circ }=372^{\circ }&\\\gamma _{5}&={\tan }^{-1}[-8.3/-6]&f_{5}=78/1.44=54\mathrm {Hz} \\&=54^{\circ }+360^{\circ }=414^{\circ }&\\\gamma _{6}&=90^{\circ }+360^{\circ }=450^{\circ }&f_{6}=74/1.44=51\mathrm {Hz} \\\gamma _{7}&={\tan }^{-1}[-9.1/7]&f_{7}=88/1.44=61\mathrm {Hz} \\&=-52^{\circ }+540^{\circ }=488^{\circ }&\\\gamma _{8}&={\tan }^{-1}[-0.4/12]&f_{8}=106/1.44=74\mathrm {Hz} \\&=-2^{\circ }+540^{\circ }=538^{\circ }&\\\gamma _{9}&={\tan }^{-1}[7.0/5]=54^{\circ }+540^{\circ }=594^{\circ }&f_{9}=92/1.44=64\mathrm {Hz} \end{aligned}} $

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): * Note: Division by 360 gives revolutions/second of the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): A(t) , equivalent to hertz.

The average instantaneous frequency is 68 Hz, the standard deviation being 11 Hz.

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