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In [[philosophy]] the term '''free will''' refers to consideration of whether an individual has the ability to make decisions or, alternatively, has only the illusion of doing so.  It is an age-old concern to separate what we can do something about, choose to do, from what we cannot.  The underlying quandary is the idea that science suggests future events are dictated to a great extent, and perhaps entirely, by past events and, inasmuch as the human body is part of the world science describes, its actions also are determined by physical laws and are not affected by human decisions. This view of events is a particular form of ''determinism'', sometimes called ''physical reductionism'',<ref name=Ney/>  and the view that determinism precludes free will is called ''incompatibilism''.<ref name=Vihvelin/> 


{{Image|High-pass amplifier Bode plot.PNG|right|300px|The Bode plot for a first-order (one-pole) [[highpass filter]]; the straight-line approximations are labeled "Bode pole"; phase varies from 90° at low frequencies (due to the contribution of the numerator, which is 90° at all
There are several ways to avoid the incompatibilst position, resulting in various ''compatibilist'' positions.<ref name=Timpe/>  One is to limit the scope of scientific description in a manner that excludes human decisions. Another is to argue that even if our actions are strictly determined by the past, it doesn’t seem that way to us, and so we have to find an approach to this issue that somehow marries our intuition of independence with the reality of its fictional nature. A third, somewhat legalistic approach, is to suggest that the ‘will’ to do something is quite different from actually doing it, so ‘free will’ can exist even though there may be no freedom of action.
frequencies) to 0° at high frequencies (where the phase contribution of the denominator is −90° and cancels the contribution of the numerator).}}
[[Image:Bode Low-Pass.PNG|350px|thumb|right|Figure 1(b): The Bode plot for a first-order (one-pole) [[lowpass filter]]; the straight-line approximations are labeled "Bode pole"; phase is 90° lower than for Figure 1(a) because the phase contribution of the numerator is 0° at all frequencies.]]
A '''Bode plot''', named after [[Hendrik Wade Bode]], is usually a combination of a Bode magnitude plot and Bode phase plot:


A '''Bode magnitude plot''' is a graph of [[logarithm|log]] magnitude versus [[frequency]], plotted with a log-frequency axis, to show the [[transfer function]] or [[frequency response]] of a [[LTI system theory|linear, time-invariant]] system. 
There is also a theological version of the dilemma. roughly, if a deity or deities, or 'fate', controls our destiny, what place is left for free will?


The magnitude axis of the Bode plot is usually expressed as [[decibel]]s, that is, 20 times the common logarithm of the amplitude gain.  With the magnitude gain being logarithmic, Bode plots make multiplication of magnitudes a simple matter of adding distances on the graph (in decibels), since
==Science does not apply==


:<math>
One approach to limiting the applicability of science to our decisions is the examination of the notion of cause and effect.  For example, [[David Hume]] suggested that science did not really deal with causality, but with the correlation of events. So, for example, lighting a match in a certain environment does not ‘’cause’’ an explosion, but is ‘’associated’’ with an explosion.  [[Immanuel Kant]] suggested that the idea of cause and effect is not a fact of nature but an interpretation put on events by the human mind, a ‘programming’ built into our brains.  Assuming this criticism to be true, there may exist classes of events that escape any attempt at cause and effect explanations.
\log(a \cdot b) = \log(a) + \log(b)\,
</math>.


A '''Bode phase plot''' is a graph of phase versus frequency, also plotted on a log-frequency axis, usually used in conjunction with the magnitude plot, to evaluate how much a frequency will be [[Phase (waves)|phase-shifted]]. For example a signal described by: ''A''sin(ω''t'') may be attenuated but also phase-shifted. If the system attenuates it by a factor ''x'' and phase shifts it by −Φ the signal out of the system will be (''A''/''x'')&nbsp;sin(ω''t''&nbsp;−&nbsp;Φ). The phase shift Φ is generally a function of frequency.  
A different way to exempt human decision from the scientific viewpoint is to note that science is a human enterprise.  It involves the human creation of theories to explain certain observations, and moreover, the observations it chooses to attempt to explain are selected, and do not encompass all experience.  For example, we choose to explain phenomena like the [[Higgs boson]] found by elaborate means like a [[hadron collider]], but don’t attempt to explain other phenomena that do not appear amenable to science at this time, often suggesting that they are beneath attention.
As time progresses, one may choose to believe that science will explain all experience, but that view must be regarded as speculation analogous to predicting the stock market on the basis of past performance.  


Phase can also be added directly from the graphical values, a fact that is mathematically clear when phase is seen as the imaginary part of the complex logarithm of a complex gain.
Although not explicitly addressing the issue of free will, it may be noted that [[Ludwig Wittgenstein]] argued that the specialized theories of science, as discussed by [[Rudolf Carnap]] for example, inevitably cover only a limited range of experience.  [[Stephen Hawking|Hawking/Mlodinow]] also noted this fact in in their [[model-dependent realism]],<ref name=Hawking/> the observation that, from the scientific viewpoint, reality is covered by a patchwork of theories that are sometimes disjoint and sometimes overlap.
{{quote|“Whatever might be the ultimate goals of some scientists, science, as it is currently practiced, depends on multiple overlapping descriptions of the world, each of which has a domain of applicability. In some cases this domain is very large, but in others quite small.”<ref name=Davies/>}}
::: —— E.B. Davies <span style="font-size:88%">''Epistemological pluralism'', p. 4</span>


In Figure 1(a), the Bode plots are shown for the one-pole [[highpass filter]] function:
Still another approach to this matter is analysis of the mind-brain connection (more generally, the [[mind-body problem]]). As suggested by Northoff,<ref name=Northoff/>  there is an observer-observation issue involved here. Observing a third-person’s mental activity is a matter for neuroscience, perhaps strictly a question of neurons and their interactions through complex networks.  But observing our own mental activity is not possible in this way – it is a matter of subjective experiences.  The suggestion has been made that ‘’complementary’’ descriptions of nature are involved, that may be simply different perspectives upon the same reality:
{{quote|“...for each individual there is ''one'' 'mental life' but ''two'' ways of knowing it: first-person knowledge and third-person knowledge. From a first-person perspective conscious experiences appear causally effective. From a third person perspective the same causal sequence can be explained in neural terms. It is not the case that the view from one perspective is right and the other wrong. These perspectives are complementary. The differences between how things appear from a first-person versus a third-person perspective has to do with differences in the ''observational arrangements'' (the means by which a subject and an external observer access the subject's mental processes).”<ref name=Velmans/> |Max Velmans: |How could conscious experiences affect brains?,  p. 11''
}}


::<math> \mathrm{T_{High}}(f) = \frac {j f/  f_1} {1 + j f/f_1} \ , </math>  
A related view is that the two descriptions may be mutually exclusive. That is, the connection between subjective experience and neuronal activity may run into a version of the measurement-observation interference noticed by [[Niels Bohr]] and by [[Erwin Schrödinger]] in the early days of quantum mechanics. (The measurement of the position of a particle caused the particle to change position in an unknown way.)  
{{quote|“...it is important to be clear about exactly what experience one wants one's subjects to introspect. Of course, explaining to subjects exactly what the experimenter wants them to experience can bring its own problems–...instructions to attend to a particular internally generated experience can easily alter both the timing and he content of that experience and even whether or not it is consciously experienced at all.”<ref name=Pockett/> |Susan Pockett |The neuroscience of movement}}


where ''f'' is the frequency in Hz, and ''f''<sub>1</sub> is the pole position in Hz, ''f''<sub>1</sub> = 100 Hz in the figure. Using the rules for [[complex number]]s, the magnitude of this function is


::<math> \mid \mathrm{T_{High}}(f) \mid = \frac { f/f_1 } { \sqrt{ 1 + (f/f_1)^2 }} \ , </math>
In any case, so far as free will is concerned, the implication of 'complementarity' is that 'free will' may be a description that is either an alternative to the scientific view, or possibly a view that can be entertained only if the scientific view is abandoned.


while the phase is:
==Science can be accommodated==
A second approach is to argue that we can accommodate our subjective notions of free will with a deterministic reality. One way to do this is to argue that although we cannot do differently, in fact we really don’t want to do differently, and so what we ‘decide’ to do always agrees with what we (in fact) have to do.  Our subjective vision of the decision process as ‘voluntary’ is just a conscious concomitant of the unconscious and predetermined move to action.


::<math> \phi_{T_{High}} = 90^\circ - \mathrm{ tan^{-1} } (f/f_1) \ . </math>
==’Will’ ''versus'' ‘action’==
There is growing evidence of the pervasive nature of subconscious thought upon our actions, and the capriciousness of consciousness,<ref name=Norretranders/> which may switch focus from a sip of coffee to the writing of a philosophical exposition without warning. There also is mounting evidence that our consciousness is greatly affected by events in the brain beyond our control. For example, [[addiction|drug addiction]] has been related to alteration of the mechanisms in the brain for [[dopamine]] production, and withdrawal from addiction requires a reprogramming of this mechanism that is more than a simple act of will. The ‘will’ to overcome addiction can become separated from the ability to execute that will.
{{quote|“Philosophers who distinguish ''freedom of action'' and ''freedom of will'' do so because our success in carrying out our ends depends in part on factors wholly beyond our control. Furthermore, there are always external constraints on the range of options we can meaningfully try to undertake. As the presence or absence of these conditions and constraints are not (usually) our responsibility, it is plausible that the central loci of our responsibility are our choices, or ‘willings’.”[Italics not in original.]<ref name=OConnor/>| Timothy O'Connor |Free Will}}
In effect, could the 'will' be a subjective perception which might operate outside the realm of scientific principle, while its execution is not?


Care must be taken that the inverse tangent is set up to return ''degrees'', not radians. On the Bode magnitude plot, decibels are used, and the plotted magnitude is:
==Theology==
The ancient Greeks held the view that the gods ''could'' intervene in the course of events, and it was possible on occasion to divine their intentions or even to change them. That view leaves a role for free will, although it can be limited in scope by the gods. A more complete restriction is the belief that the gods are omniscient and have perfect foreknowledge of events, which obviously includes human decisions. This view leads to the belief that, while the gods know what we will choose, humans do not, and are faced therefore with playing the role of deciding our actions, even though they are scripted, a view contradicted by Cassius in arguing with Brutus, a Stoic:
{{quote| “The fault, dear Brutus, is not in our stars, But in ourselves, that we are underlings.”|spoken by Cassius| Julius Caesar (I, ii, 140-141)}}
The [[Stoicism|Stoics]] wrestled with this problem, and one argument for compatibility took the view that although the gods controlled matters, what they did was understandable using human intellect. Hence, when fate presented us with an issue, there was a duty to sort through a decision, and assent to it (a ''responsibility''), a sequence demanded by our natures as rational beings.<ref name=Bobzien/>


:<math>20\ \mathrm{log_{10}} \mid \mathrm{T_{High}}(f) \mid \ =20\  \mathrm{log_{10}} \left( f/f_1 \right)</math>
In [http://www.iep.utm.edu/chrysipp/ Chrysippus of Soli's] view (an apologist for Stoicism), ''fate'' precipitates an event, but our nature determines its course, in the same way that bumping a cylinder or a cone causes it to move, but it rolls or it spins according to its nature.<ref name=Bobzien2/> The actual course of events depends upon the nature of the individual, who therefore bears a personal responsibility for the resulting events. It is not clear whether the individual is thought to have any control over their nature, or even whether this question has any bearing upon their responsibility.<ref name=Bobzien3/>
:::::::&emsp; <math>\ -20  \  \mathrm{log_{10}} \left( \sqrt{ 1 + (f/f_1)^2 }\right) \ . </math>
In Figure 1(b), the Bode plots are shown for the one-pole [[lowpass filter]] function:


::<math> \mathrm{ T_{Low}} (f) = \frac {1} {1 + j f/f_1} \ . </math>
==References==
{{reflist|refs=
<ref name=Bobzien>
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See §6.3.3 ''The cylinder and cone analogy'',  pp. 258 ''ff''.  
</ref>


Also shown in Figure 1(a) and 1(b) are the straight-line approximations to the Bode plots that are used in hand analysis, and described later.
<ref name=Bobzien2>
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See in particular pp. 386 ''ff''.  
</ref>


The magnitude and phase Bode plots can seldom be changed independently of each other — changing the amplitude response of the system will most likely change the phase characteristics and vice versa. For minimum-phase systems the phase and amplitude characteristics can be obtained from each other with the use of the [[Hilbert transform]].
<ref name=Bobzien3>
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See in particular p. 255.  
</ref>


If the transfer function is a [[rational function]] with real poles and zeros, then the Bode plot can be approximated with straight lines. These asymptotic approximations are called '''straight line Bode plots''' or '''uncorrected Bode plots''' and are useful because they can be drawn by hand following a few simple rules.  Simple plots can even be predicted without drawing them.


The approximation can be taken further by ''correcting'' the value at each cutoff frequency. The plot is then called a '''corrected Bode plot'''.
<ref name=Davies>
{{cite web |title=Epistemological pluralism |author=E Brian Davies |url=http://philsci-archive.pitt.edu/3083/1/EP3single.doc |work=PhilSci Archive |year=2006 }}
</ref>


==Rules for hand-made Bode plot==


The main idea about Bode plots is that one can think of the log of a function in the form:
<ref name=Hawking>
:<math> f(x) = A \prod (x + c_n)^{a_n} </math>
{{cite book |author=Hawking SW, Mlodinow L. |title=The Grand Design |isbn=0553805371 |url= http://www.amazon.com/Grand-Design-Stephen-Hawking/dp/0553805371#reader_0553805371 |pages=pp. 42-43 |chapter=Chapter 3: What is reality?|year=2010|publisher=Bantam Books}}
</ref>


as a sum of the logs of its [[Pole (complex analysis)|poles]] and [[Zero (complex analysis)|zeros]]:
<ref name=Norretranders>
:<math> \log(f(x)) = \log(A) + \sum a_n \log(x + c_n). </math>
{{cite book |url=http://www.google.com/search?tbo=p&tbm=bks&q=consciousness%2Bplays%2Ba%2Bsmaller%2Brole%2Bin%2Bhuman%2Blife+intitle:User+intitle:illusion&num=10 |title=The user illusion: Cutting consciousness down to size |quote=Consciousness plays a far smaller role in human life than Western culture has tended to believe |author=Tor Nørretranders |isbn=0140230122 |chapter=Preface |pages=p. ''ix'' |publisher=Penguin Books |year=1998 |edition=Jonathan Sydenham translation of ''Maerk verden'' 1991 ed }}
</ref>


This idea is used explicitly in the method for drawing phase diagrams. The method for drawing amplitude plots implicitly uses this idea, but since the log of the amplitude of each pole or zero always starts at zero and only has one asymptote change (the straight lines), the method can be simplified.
<ref name=Ney>
{{cite web |author=Alyssa Ney |title=Reductionism |work=Internet Encyclopedia of Philosophy |date= November 10, 2008 |url=http://www.iep.utm.edu/red-ism/}}
</ref>


=== Straight-line amplitude plot===
<ref name=Northoff>
Amplitude decibels is usually done using the <math> 20 \log_{10}(X)</math> version. Given a transfer function in the form
A rather extended discussion is provided in {{cite book |title=Philosophy of the Brain: The Brain Problem |author=Georg Northoff |url=http://books.google.com/books?id=r0Bf3lLys6AC&printsec=frontcover |publisher=John Benjamins Publishing |isbn=1588114171 |year=2004 |edition=Volume 52 of Advances in Consciousness Research}}
:<math> H(s) = A \prod \frac{(s + x_n)^{a_n}}{(s + y_n)^{b_n}} </math>
</ref>
:: where <math>x_n</math> and <math>y_n</math> are constants, <math>s = j\omega</math>, <math>a_n</math>, <math>b_n</math> > 0, and H is the transfer function:
 
* at every value of s where <math>\omega = x_n</math> (a zero), '''increase''' the slope of the line by <math>20 \cdot a_n dB</math> per decade.
* at every value of s where <math>\omega = y_n</math> (a pole), '''decrease''' the slope of the line by <math>20 \cdot b_n dB </math> per decade.
* The initial value of the graph depends on the boundaries. The initial point is found by putting the initial angular frequency ''ω'' into the function and finding |H(jω)|.
* The initial slope of the function at the initial value depends on the number and order of zeros and poles that are at values below the initial value, and are found using the first two rules.
 
To handle irreducible 2nd order polynomials, <math> ax^2 + bx + c \ </math> can, in many cases, be approximated as <math> (\sqrt{a}x + \sqrt{c})^2 </math>.
 
Note that zeros and poles happen when ω is ''equal to'' a certain <math>x_n</math> or <math>y_n</math>. This is because the function in question is the magnitude of H(jω), and since it is a complex function, <math>|H(j\omega)| = \sqrt{H \cdot H^* } </math>. Thus at any place where there is a zero or pole involving the term <math>(s + x_n) </math>, the magnitude of that term is <math>\sqrt{(x_n + j\omega) \cdot (x_n - j\omega)}= \sqrt{x_n^2+\omega^2}</math>.
 
=== Corrected amplitude plot===
 
To correct a straight-line amplitude plot:
 
* at every zero, put a point <math>3 \cdot a_n\ \mathrm{dB}</math>  '''above''' the line,
* at every pole, put a point <math>3 \cdot b_n\ \mathrm{dB}</math> '''below''' the line,
* draw a smooth line through those points using the straight lines as asymptotes (lines which the curve approaches).
 
Note that this correction method does not incorporate how to handle complex values of <math> x_n </math> or <math> y_n </math>. In the case of an irreducible polynomial, the best way to correct the plot is to actually calculate the magnitude of the transfer function at the pole or zero corresponding to the irreducible polynomial, and put that dot over or under the line at that pole or zero.
 
=== Straight-line phase plot ===
 
Given a transfer function in the same form as above:
:<math> H(s) = A \prod \frac{(s + x_n)^{a_n}}{(s + y_n)^{b_n}} </math>
the idea is to draw separate plots for each pole and zero, then add them up. The actual phase curve is given by <math>-\mathrm{arctan}(\mathrm{Im}[H(s)] / \mathrm{Re}[H(s)])</math>.
 
To draw the phase plot, for '''each''' pole and zero:
 
* if A is positive, start line (with zero slope) at 0 degrees
* if A is negative, start line (with zero slope) at 180 degrees
* at every <math> \omega = x_n </math> (for stable zeros – <math>Re(z) < 0</math>), '''increase''' the slope by <math>45 \cdot a_n</math> degrees per decade, beginning one decade before <math> \omega = x_n </math> (i.e. <math> \frac{x_n}{10} </math>)
* at every <math> \omega = y_n </math> (for stable poles – <math>Re(p) < 0</math>), '''decrease''' the slope by <math>45 \cdot b_n</math> degrees per decade, beginning one decade before <math> \omega = y_n </math> (i.e.  <math> \frac{y_n}{10} </math>)
* "unstable" (right half plane) poles and zeros (<math>Re(s) > 0</math>) have opposite behavior
* flatten the slope again when the phase has changed by <math>90 \cdot a_n</math> degrees (for a zero) or <math>90 \cdot b_n</math> degrees (for a pole),
* After plotting one line for each pole or zero, add the lines together to obtain the final phase plot; that is, the final phase plot is the superposition of each earlier phase plot.
 
==Example==
 
A passive (unity pass band gain) [[lowpass]] [[RC circuit|RC filter]], for instance has the following [[transfer function]] expressed in the [[frequency domain]]:
 
:<math>
H(jf) = \frac{1}{1+j2\pi f R C}
</math>
 
From the transfer function it can be determined that the [[cutoff frequency]] point ''f''<sub>c</sub> (in [[hertz]]) is at the frequency
:<math>
f_\mathrm{c} = {1 \over {2\pi RC}}
</math>
:or (equivalently) at
:<math>
\omega_\mathrm{c} = {1 \over {RC}}
</math> where <math>\omega_\mathrm{c}=2\pi f_\mathrm{c}</math> is the angular cutoff frequency in radians per second.


The transfer function in terms of the angular frequencies becomes:
<ref name=OConnor>
:<math>
{{cite web |title=&thinsp;Free Will |date=Oct 29, 2010 |author=O'Connor, Timothy |url=http://plato.stanford.edu/archives/sum2011/entries/freewill |work=The Stanford Encyclopedia of Philosophy (Summer 2011 Edition) |editor=Edward N. Zalta, ed.}}
H(j\omega)  = {1 \over 1+j{\omega \over {{\omega_\mathrm{c}}}}}
</math>
The above equation is the normalized form of the transfer function. The Bode plot is shown in Figure 1(b) above, and construction of the straight-line approximation is discussed next.
 
===Magnitude plot===
 
The magnitude (in [[decibel]]s) of the transfer function above, (normalized and converted to angular frequency form), given by the decibel gain expression <math>A_\mathrm{vdB}</math>:
:<math>
A_\mathrm{vdB} \ = \ 20\ \log|H(j\omega)| \ = \ 20\  \log {1 \over \left|1+j{\omega \over {{\omega_\mathrm{c}}}}\right|} </math>
:::<math>= \ - 20\  \log \left|1+j{\omega \over {{\omega_\mathrm{c}}}}\right| </math> &ensp; <math> = \ -10\  \log{\left[1 + \frac{\omega^2}{\omega_\mathrm{c}^2}\right]}
</math>
 
when plotted versus input frequency <math>\omega</math> on a logarithmic scale, can be approximated by two lines and it forms the asymptotic (approximate) magnitude Bode plot of the transfer function:
* for angular frequencies below <math>\omega_\mathrm{c}</math> it is a horizontal line at 0 dB since at low frequencies the <math>{\omega \over {\omega_\mathrm{c}}}</math> term is small and can be neglected, making the decibel gain equation above equal to zero,
* for angular frequencies above <math>\omega_\mathrm{c}</math> it is a line with a slope of −20 dB per decade since at high frequencies the <math>{\omega \over {\omega_\mathrm{c}}}</math> term dominates and the decibel gain expression above simplifies to <math>-20 \log {\omega \over {\omega_\mathrm{c}}}</math> which is a straight line with a slope of -20 dB per decade.
 
These two lines meet at the [[corner frequency]]. From the plot, it can be seen that for frequencies well below the corner frequency, the circuit has an attenuation of 0 dB, corresponding to a unity pass band gain, i.e. the amplitude of the filter output equals the amplitude of the input. Frequencies above the corner frequency are attenuated – the higher the frequency, the higher the [[attenuation]].
[[Image:Bode Low Pass Magnitude Plot.PNG|thumbnail|400px|Figure 2: Bode magnitude plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots]]
[[Image:Bode Low Pass Phase Plot.PNG|thumbnail|400px|Figure 3: Bode phase plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots]]
[[Image:Bode Pole-Zero Magnitude Plot.PNG|thumbnail|400px|Figure 4: Bode magnitude plot for pole-zero combination; the location of the zero is ten times higher than in Figures 2&3; curves labeled "Bode" are the straight-line Bode plots]]
[[Image:Bode Pole-Zero Phase Plot.PNG|thumbnail|400px|Figure 5: Bode phase plot for pole-zero combination; the location of the zero is ten times higher than in Figures 2&3; curves labeled "Bode" are the straight-line Bode plots]]
 
===Phase plot===
The phase Bode plot is obtained by plotting the phase angle of the transfer function given by:
<math>
\phi = -\tan^{-1}{\omega \over {\omega_\mathrm{c}}}
</math>
versus <math>\omega</math>, where <math>\omega</math> and <math>\omega_\mathrm{c}</math> are the input and cutoff angular frequencies respectively. For input frequencies much lower than corner, the ratio <math> {\omega \over {\omega_\mathrm{c}}}</math> is small and therefore the phase angle is close to zero. As the ratio increases the absolute value of the phase increases and becomes –45 degrees when <math>\omega =\omega_\mathrm{c}</math>. As the ratio increases for input frequencies much greater than the corner frequency, the phase angle asymptotically approaches -90 degrees. The frequency scale for the phase plot is logarithmic.
 
===Normalized plot===
The horizontal frequency axis, in both the magnitude and phase plots, can be replaced by the normalized (nondimensional) frequency ratio <math>{\omega \over {\omega_\mathrm{c}}}</math>. In such a case the plot is said to be normalized and units of the frequencies are no longer used since all input frequencies are now expressed as multiples of the cutoff frequency <math>\omega_\mathrm{c}</math>.
 
==An example with pole and zero==
Figures 2-5 further illustrate construction of Bode plots. This example with both a pole and a zero shows how to use superposition. To begin, the components are presented separately.
 
Figure 2 shows the Bode magnitude plot for a zero and a low-pass pole, and compares the two with the Bode straight line plots. The straight-line plots are horizontal up to the pole (zero) location and then drop (rise) at 20 dB/decade. The second Figure 3 does the same for the phase. The phase plots are horizontal up to a frequency a factor of ten below the pole (zero) location and then drop (rise) at 45°/decade until the frequency is ten times higher than the pole (zero) location. The plots then are again horizontal at higher frequencies at a final, total phase change of 90°.
 
Figure 4 and Figure 5 show how superposition (simple addition) of a pole and zero plot is done. The Bode straight line plots again are compared with the exact plots. The zero has been moved to higher frequency than the pole to make a more interesting example. Notice in Figure 4 that the 20 dB/decade drop of the pole is arrested by the 20 dB/decade rise of the zero resulting in a horizontal magnitude plot for  frequencies above the zero location. Notice in Figure 5 in the phase plot that the straight-line approximation is pretty approximate in the region where both pole and zero affect the phase. Notice also in Figure 5 that the range of frequencies where the phase changes in the straight line plot is limited to frequencies a factor of ten above and below the pole (zero) location. Where the phase of the pole and the zero both are present, the straight-line phase plot is horizontal because the 45°/decade drop of the pole is arrested by the overlapping 45°/decade rise of the zero in the limited range of frequencies where both are active contributors to the phase.
 
==Gain margin and phase margin==
Bode plots are used to assess the stability of negative feedback amplifiers by finding the gain and [[phase margin]]s of an amplifier. The notion of gain and phase margin is based upon the gain expression for a [[negative feedback amplifier]] given by
 
::<math> A_{FB} = \frac {A_{OL}} {1 + \beta A_{OL}} \ , </math>
 
where A<sub>FB</sub> is the gain of the amplifier with feedback (the '''closed-loop gain'''), β is the '''feedback factor''' and ''A''<sub>OL</sub> is the gain without feedback (the '''open-loop gain'''). The gain ''A''<sub>OL</sub> is a complex function of frequency, with both magnitude and phase.<ref name=note1/> Examination of this relation shows the possibility of infinite gain (interpreted as instability) if the product β''A''<sub>OL</sub> = −1. (That is, the magnitude of β''A''<sub>OL</sub> is unity and its phase is −180°, the so-called '''Barkhausen  criteria'''). Bode plots are used to determine just how close an amplifier comes to satisfying this condition.
 
Key to this determination are two frequencies. The first, labeled here as ''f''<sub>180</sub>, is the frequency where the open-loop gain flips sign. The second, labeled here ''f''<sub>0dB</sub>, is the frequency where the magnitude of the product | β ''A''<sub>OL</sub> | = 1 (in dB, magnitude 1 is 0 dB). That is, frequency ''f''<sub>180</sub> is determined by the condition:
 
:::<math> \beta A_{OL} \left( f_{180} \right) = - | \beta A_{OL} \left( f_{180} \right)| = - | \beta A_{OL}|_{180} \ , </math>
 
where vertical bars denote the magnitude of a complex number (for example, | a + j b | = [ a<sup>2</sup> + b<sup>2</sup>]<sup>1/2</sup> ), and frequency ''f''<sub>0dB</sub> is determined by the condition:
 
:::<math>| \beta A_{OL} \left( f_{0dB} \right) | = 1 \ . </math>
 
One measure of proximity to instability is the '''gain margin'''. The Bode phase plot locates the frequency where the phase of β''A''<sub>OL</sub> reaches −180°, denoted here as frequency ''f''<sub>180</sub>. Using this frequency, the Bode magnitude plot finds the magnitude of β''A''<sub>OL</sub>. If |β''A''<sub>OL</sub>|<sub>180</sub> = 1, the amplifier is unstable, as mentioned. If |β''A''<sub>OL</sub>|<sub>180</sub> < 1, instability does not occur, and the separation in dB of the magnitude of |β''A''<sub>OL</sub>|<sub>180</sub> from |β''A''<sub>OL</sub>| = 1 is called the ''gain margin''. Because a magnitude of one is 0 dB, the gain margin is simply one of the equivalent forms: 20 log<sub>10</sub>( |β''A''<sub>OL</sub>|<sub>180</sub>) = 20 log<sub>10</sub>( |''A''<sub>OL</sub>|<sub>180</sub>) − 20 log<sub>10</sub>( 1 / β ).
 
Another equivalent measure of proximity to instability is the '''[[phase margin]]'''. The Bode magnitude plot locates the frequency where the magnitude of |β''A''<sub>OL</sub>| reaches unity, denoted here as frequency ''f''<sub>0dB</sub>. Using this frequency, the Bode phase plot finds the phase of β''A''<sub>OL</sub>. If the phase of β''A''<sub>OL</sub>( ''f''</sub><sub>0dB</sub>) > −180°,  the instability condition cannot be met at any frequency (because its magnitude is going to be < 1 when ''f = f''<sub>180</sub>), and the distance of the phase at ''f''<sub>0dB</sub> in degrees above −180° is called the ''phase margin''.
 
If a simple ''yes'' or ''no'' on the stability issue is all that is needed, the amplifier is stable if ''f''<sub>0dB</sub> < ''f''<sub>180</sub>. This criterion is sufficient to predict stability only for amplifiers satisfying some restrictions on their pole and zero positions ([[minimum phase]] systems). Although these restrictions usually are met, if they are not another method must be used, such as the [[Nyquist plot]].<ref name=Lee/><ref name=Levine/>
[[Image:Magnitude of feedback amplifier.PNG|thumbnail|380px|Figure 6: Gain of feedback amplifier ''A''<sub>FB</sub> in dB and corresponding open-loop amplifier ''A''<sub>OL</sub>. The gain margin in this amplifier is nearly zero because &#124; β''A''<sub>OL</sub>&#124; = 1 occurs at almost ''f'' = ''f''<sub>180°</sub>.]]
[[Image:Phase of feedback amplifier.PNG|thumbnail|380px|Figure 7: Phase of feedback amplifier ''°A''<sub>FB</sub> in degrees and corresponding open-loop amplifier ''°A''<sub>OL</sub>. The phase margin in this amplifier is nearly zero because the phase-flip occurs at almost the unity gain frequency ''f'' = ''f''<sub>0dB</sub> where &#124; β''A''<sub>OL</sub>&#124; = 1.]]
===Examples using Bode plots===
Figures 6 and 7 illustrate the gain behavior and terminology. For a three-pole amplifier, Figure 6 compares the Bode plot for the gain without feedback (the ''open-loop'' gain) ''A''<sub>OL</sub> with the gain with feedback ''A''<sub>FB</sub> (the ''closed-loop'' gain). See [[negative feedback amplifier]] for more detail.  
 
Because the open-loop gain ''A''<sub>OL</sub> is plotted and not the product β ''A''<sub>OL</sub>, the condition ''A''<sub>OL</sub> = 1 / β decides ''f''<sub>0dB</sub>. The feedback gain at low frequencies and for large ''A''<sub>OL</sub> is ''A''<sub>FB</sub> &asymp; 1 / β (look at the formula for the feedback gain at the beginning of this section for the case of large gain ''A''<sub>OL</sub>), so an equivalent way to find ''f''<sub>0dB</sub> is to look where the feedback gain intersects the open-loop gain. (Frequency ''f''<sub>0dB</sub> is needed later to find the phase margin.)
 
Near this crossover of the two gains at ''f''<sub>0dB</sub>, the Barkhausen criteria are almost satisfied in this example, and the feedback amplifier exhibits a massive peak in gain (it would be infinity if β ''A''<sub>OL</sub> = −1). Beyond the unity gain frequency ''f''<sub>0dB</sub>, the open-loop gain is sufficiently small that ''A''<sub>FB</sub> ≈ ''A''<sub>OL</sub> (examine the formula at the beginning of this section for the case of small ''A''<sub>OL</sub>).
 
Figure 7 shows the corresponding phase comparison: the phase of the feedback amplifier is nearly zero out to the frequency ''f''<sub>180</sub> where the open-loop gain has a phase of −180°. In this vicinity, the phase of the feedback amplifier plunges abruptly downward to become almost the same as the phase of the open-loop amplifier. (Recall, ''A''<sub>FB</sub> ≈ ''A''<sub>OL</sub> for small ''A''<sub>OL</sub>.)
 
Comparing the labeled points in Figure 6 and Figure 7, it is seen that the unity gain frequency ''f''<sub>0dB</sub> and the phase-flip frequency ''f''<sub>180</sub> are very nearly equal in this amplifier, ''f''<sub>180</sub> &asymp; ''f''<sub>0dB</sub> &asymp; 3.332 kHz, which means the gain margin and phase margin are nearly zero. The amplifier is borderline stable.
[[Image:Gain Margin.PNG|thumbnail|380px|Figure 8: Gain of feedback amplifier ''A''<sub>FB</sub> in dB and corresponding open-loop amplifier ''A''<sub>OL</sub>. The gain margin in this amplifier is 19 dB.]]
[[Image:Phase Margin.PNG|thumbnail|380px|Figure 9: Phase of feedback amplifier ''A''<sub>FB</sub> in degrees and corresponding open-loop amplifier ''A''<sub>OL</sub>. The phase margin in this amplifier is 45°.]]
Figures 8 and 9 illustrate the gain margin and phase margin for a different amount of feedback β. The feedback factor is chosen smaller than in Figure 6 or 7, moving the the condition | β ''A''<sub>OL</sub> | = 1 to lower frequency.
 
Figure 8 shows the gain plot. From Figure 8, the intersection of 1 / β and ''A''<sub>OL</sub> occurs at  ''f''<sub>0dB</sub> = 1 kHz. Notice that the peak in the gain ''A''<sub>FB</sub> near ''f''<sub>0dB</sub> is almost gone.<ref name=note2/><ref name=Sansen/>
Figure 9 is the phase plot. Using the value of ''f''<sub>0dB</sub> = 1 kHz found above from the magnitude plot of Figure 8, the open-loop phase at ''f''<sub>0dB</sub> is −135°, which is a phase margin of 45° above −180°.
 
Using Figure 9, for a phase of −180° the value of ''f''<sub>180</sub> = 3.332 kHz (the same result as found earlier, of course<ref name=note3/>). The open-loop gain from Figure 8 at ''f''<sub>180</sub> is 58 dB, and 1 / β = 77 dB, so the gain margin is 19 dB.
 
As an aside, it should be noted that stability is not the sole criterion for amplifier response, and in many applications a more stringent demand than stability is good [[Step_response#Step_response_of_feedback_amplifiers|step response]]. As a [[rule of thumb]], good step response requires a phase margin of at least 45°, and often a margin of over 70° is advocated, particularly where component variation due to manufacturing tolerances is an issue.<ref name=Sansen2/> See also the discussion of phase margin in the [[Step_response#Phase_margin|step response]] article.
 
==References and notes==
{{reflist|refs=
<ref name=note1>
Ordinarily, as frequency increases the magnitude of the gain drops and the phase becomes more negative, although these are only trends and may be reversed in particular frequency ranges. Unusual gain behavior can render the concepts of gain and phase margin inapplicable. Then other methods such as the [[Nyquist plot]] have to be used to assess stability.
</ref>
</ref>


<ref name=Lee>
<ref name=Pockett>
{{cite book  
{{cite book|title=&thinsp;Does Consciousness Cause Behavior? |chapter=The neuroscience of movement |author=Susan Pockett |url=http://books.google.com/books?id=G5CaTnNksgkC&pg=PA19&lpg=PA19 |pages= p. 19 |editor=Susan Pockett, WP Banks, Shaun Gallagher, eds.  |publisher=MIT Press |date =2009 |isbn=0262512572}}
|author=Thomas H. Lee
|title=The design of CMOS radio-frequency integrated circuits
|page=§14.6 pp. 451-453
|year= 2004
|edition=Second Edition
|publisher=Cambridge University Press
|location=Cambridge UK
|isbn=0-521-83539-9
|url=http://worldcat.org/isbn/0-521-83539-9}}
</ref>
</ref>


<ref name=Levine>
<ref name=Timpe>
{{cite book
{{cite web |author=Kevin Timpe |title=Free will |work=Internet Encyclopedia of Philosophy |date= March 31, 2006 |url=http://www.iep.utm.edu/freewill/#H5}}
|author=William S Levine
|title=The control handbook: the electrical engineering handbook series
|page=§10.1 p. 163
|year= 1996
|edition=Second Edition
|publisher=CRC Press/IEEE Press
|location=Boca Raton FL
|isbn=0849385709
|url=http://books.google.com/books?id=2WQP5JGaJOgC&pg=RA1-PA163&lpg=RA1-PA163&dq=stability+%22minimum+phase%22&source=web&ots=P3fFTcyfzM&sig=ad5DJ7EvVm6In_zhI0MlF_6vHDA}}
</ref>
 
<ref name=note2>The critical amount of feedback where the peak in the gain ''just'' disappears altogether is the ''maximally flat'' or [[Butterworth_filter#Maximal_flatness|Butterworth]] design.
</ref>
</ref>


<ref name=note3>
The frequency where the open-loop gain flips sign ''f''<sub>180</sub> does not change with a change in feedback factor; it is a property of the open-loop gain. The value of the gain at ''f''<sub>180</sub> also does not change with a change in β. Therefore, we could use the previous values from Figures 6 and 7. However, for clarity the procedure is described using only Figures 8 and 9.
</ref>


<ref name=Sansen>
<ref name=Velmans>
{{cite book
{{cite journal |journal=Journal of Consciousness Studies |volume=9 |issue=11 |year=2002 |pages=pp. 2-29 |author=Max Velmans  |title=How Could Conscious Experiences Affect Brains? |url=http://cogprints.org/2750/ |year=2002}}
|author=Willy M C Sansen
|title=Analog design essentials
|page=§0517-§0527 pp. 157-163
|year= 2006
|publisher=Springer
|location=Dordrecht, The Netherlands
|isbn=0-387-25746-2
|url=http://worldcat.org/isbn/0-387-25746-2}}
</ref>
</ref>


 
<ref name=Vihvelin>
<ref name=Sansen2>
{{cite web |author=Kadri Vihvelin |title=&thinsp;Arguments for Incompatibilism |work=The Stanford Encyclopedia of Philosophy (Spring 2011 Edition) |editor=Edward N. Zalta, ed. |url= http://plato.stanford.edu/archives/spr2011/entries/incompatibilism-arguments/ |date=Mar 1, 2011}}
{{cite book
|author=Willy M C Sansen
|title=§0526 p. 162
|isbn=0-387-25746-2
|url=http://worldcat.org/isbn/0-387-25746-2}}
</ref>
</ref>


}}
}}

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In philosophy the term free will refers to consideration of whether an individual has the ability to make decisions or, alternatively, has only the illusion of doing so. It is an age-old concern to separate what we can do something about, choose to do, from what we cannot. The underlying quandary is the idea that science suggests future events are dictated to a great extent, and perhaps entirely, by past events and, inasmuch as the human body is part of the world science describes, its actions also are determined by physical laws and are not affected by human decisions. This view of events is a particular form of determinism, sometimes called physical reductionism,[1] and the view that determinism precludes free will is called incompatibilism.[2]

There are several ways to avoid the incompatibilst position, resulting in various compatibilist positions.[3] One is to limit the scope of scientific description in a manner that excludes human decisions. Another is to argue that even if our actions are strictly determined by the past, it doesn’t seem that way to us, and so we have to find an approach to this issue that somehow marries our intuition of independence with the reality of its fictional nature. A third, somewhat legalistic approach, is to suggest that the ‘will’ to do something is quite different from actually doing it, so ‘free will’ can exist even though there may be no freedom of action.

There is also a theological version of the dilemma. roughly, if a deity or deities, or 'fate', controls our destiny, what place is left for free will?

Science does not apply

One approach to limiting the applicability of science to our decisions is the examination of the notion of cause and effect. For example, David Hume suggested that science did not really deal with causality, but with the correlation of events. So, for example, lighting a match in a certain environment does not ‘’cause’’ an explosion, but is ‘’associated’’ with an explosion. Immanuel Kant suggested that the idea of cause and effect is not a fact of nature but an interpretation put on events by the human mind, a ‘programming’ built into our brains. Assuming this criticism to be true, there may exist classes of events that escape any attempt at cause and effect explanations.

A different way to exempt human decision from the scientific viewpoint is to note that science is a human enterprise. It involves the human creation of theories to explain certain observations, and moreover, the observations it chooses to attempt to explain are selected, and do not encompass all experience. For example, we choose to explain phenomena like the Higgs boson found by elaborate means like a hadron collider, but don’t attempt to explain other phenomena that do not appear amenable to science at this time, often suggesting that they are beneath attention. As time progresses, one may choose to believe that science will explain all experience, but that view must be regarded as speculation analogous to predicting the stock market on the basis of past performance.

Although not explicitly addressing the issue of free will, it may be noted that Ludwig Wittgenstein argued that the specialized theories of science, as discussed by Rudolf Carnap for example, inevitably cover only a limited range of experience. Hawking/Mlodinow also noted this fact in in their model-dependent realism,[4] the observation that, from the scientific viewpoint, reality is covered by a patchwork of theories that are sometimes disjoint and sometimes overlap.

‘ “Whatever might be the ultimate goals of some scientists, science, as it is currently practiced, depends on multiple overlapping descriptions of the world, each of which has a domain of applicability. In some cases this domain is very large, but in others quite small.”[5] ’
—— E.B. Davies Epistemological pluralism, p. 4

Still another approach to this matter is analysis of the mind-brain connection (more generally, the mind-body problem). As suggested by Northoff,[6] there is an observer-observation issue involved here. Observing a third-person’s mental activity is a matter for neuroscience, perhaps strictly a question of neurons and their interactions through complex networks. But observing our own mental activity is not possible in this way – it is a matter of subjective experiences. The suggestion has been made that ‘’complementary’’ descriptions of nature are involved, that may be simply different perspectives upon the same reality:

‘ “...for each individual there is one 'mental life' but two ways of knowing it: first-person knowledge and third-person knowledge. From a first-person perspective conscious experiences appear causally effective. From a third person perspective the same causal sequence can be explained in neural terms. It is not the case that the view from one perspective is right and the other wrong. These perspectives are complementary. The differences between how things appear from a first-person versus a third-person perspective has to do with differences in the observational arrangements (the means by which a subject and an external observer access the subject's mental processes).”[7] ’

—Max Velmans: , How could conscious experiences affect brains?, p. 11

A related view is that the two descriptions may be mutually exclusive. That is, the connection between subjective experience and neuronal activity may run into a version of the measurement-observation interference noticed by Niels Bohr and by Erwin Schrödinger in the early days of quantum mechanics. (The measurement of the position of a particle caused the particle to change position in an unknown way.)

‘ “...it is important to be clear about exactly what experience one wants one's subjects to introspect. Of course, explaining to subjects exactly what the experimenter wants them to experience can bring its own problems–...instructions to attend to a particular internally generated experience can easily alter both the timing and he content of that experience and even whether or not it is consciously experienced at all.”[8] ’

—Susan Pockett , The neuroscience of movement


In any case, so far as free will is concerned, the implication of 'complementarity' is that 'free will' may be a description that is either an alternative to the scientific view, or possibly a view that can be entertained only if the scientific view is abandoned.

Science can be accommodated

A second approach is to argue that we can accommodate our subjective notions of free will with a deterministic reality. One way to do this is to argue that although we cannot do differently, in fact we really don’t want to do differently, and so what we ‘decide’ to do always agrees with what we (in fact) have to do. Our subjective vision of the decision process as ‘voluntary’ is just a conscious concomitant of the unconscious and predetermined move to action.

’Will’ versus ‘action’

There is growing evidence of the pervasive nature of subconscious thought upon our actions, and the capriciousness of consciousness,[9] which may switch focus from a sip of coffee to the writing of a philosophical exposition without warning. There also is mounting evidence that our consciousness is greatly affected by events in the brain beyond our control. For example, drug addiction has been related to alteration of the mechanisms in the brain for dopamine production, and withdrawal from addiction requires a reprogramming of this mechanism that is more than a simple act of will. The ‘will’ to overcome addiction can become separated from the ability to execute that will.

‘ “Philosophers who distinguish freedom of action and freedom of will do so because our success in carrying out our ends depends in part on factors wholly beyond our control. Furthermore, there are always external constraints on the range of options we can meaningfully try to undertake. As the presence or absence of these conditions and constraints are not (usually) our responsibility, it is plausible that the central loci of our responsibility are our choices, or ‘willings’.”[Italics not in original.][10] ’

— Timothy O'Connor , Free Will

In effect, could the 'will' be a subjective perception which might operate outside the realm of scientific principle, while its execution is not?

Theology

The ancient Greeks held the view that the gods could intervene in the course of events, and it was possible on occasion to divine their intentions or even to change them. That view leaves a role for free will, although it can be limited in scope by the gods. A more complete restriction is the belief that the gods are omniscient and have perfect foreknowledge of events, which obviously includes human decisions. This view leads to the belief that, while the gods know what we will choose, humans do not, and are faced therefore with playing the role of deciding our actions, even though they are scripted, a view contradicted by Cassius in arguing with Brutus, a Stoic:

‘ “The fault, dear Brutus, is not in our stars, But in ourselves, that we are underlings.” ’

—spoken by Cassius, Julius Caesar (I, ii, 140-141)

The Stoics wrestled with this problem, and one argument for compatibility took the view that although the gods controlled matters, what they did was understandable using human intellect. Hence, when fate presented us with an issue, there was a duty to sort through a decision, and assent to it (a responsibility), a sequence demanded by our natures as rational beings.[11]

In Chrysippus of Soli's view (an apologist for Stoicism), fate precipitates an event, but our nature determines its course, in the same way that bumping a cylinder or a cone causes it to move, but it rolls or it spins according to its nature.[12] The actual course of events depends upon the nature of the individual, who therefore bears a personal responsibility for the resulting events. It is not clear whether the individual is thought to have any control over their nature, or even whether this question has any bearing upon their responsibility.[13]

References

  1. ↑ Alyssa Ney (November 10, 2008). Reductionism. Internet Encyclopedia of Philosophy.
  2. ↑ Kadri Vihvelin (Mar 1, 2011). Edward N. Zalta, ed.: Arguments for Incompatibilism. The Stanford Encyclopedia of Philosophy (Spring 2011 Edition).
  3. ↑ Kevin Timpe (March 31, 2006). Free will. Internet Encyclopedia of Philosophy.
  4. ↑ Hawking SW, Mlodinow L. (2010). “Chapter 3: What is reality?”, The Grand Design. Bantam Books, pp. 42-43. ISBN 0553805371. 
  5. ↑ E Brian Davies (2006). Epistemological pluralism. PhilSci Archive.
  6. ↑ A rather extended discussion is provided in Georg Northoff (2004). Philosophy of the Brain: The Brain Problem, Volume 52 of Advances in Consciousness Research. John Benjamins Publishing. ISBN 1588114171. 
  7. ↑ Max Velmans (2002). "How Could Conscious Experiences Affect Brains?". Journal of Consciousness Studies 9 (11): pp. 2-29.
  8. ↑ Susan Pockett (2009). “The neuroscience of movement”, Susan Pockett, WP Banks, Shaun Gallagher, eds.:  Does Consciousness Cause Behavior?. MIT Press, p. 19. ISBN 0262512572. 
  9. ↑ Tor Nørretranders (1998). “Preface”, The user illusion: Cutting consciousness down to size, Jonathan Sydenham translation of Maerk verden 1991 ed. Penguin Books, p. ix. ISBN 0140230122. “Consciousness plays a far smaller role in human life than Western culture has tended to believe” 
  10. ↑ O'Connor, Timothy (Oct 29, 2010). Edward N. Zalta, ed.: Free Will. The Stanford Encyclopedia of Philosophy (Summer 2011 Edition).
  11. ↑ Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See §6.3.3 The cylinder and cone analogy, pp. 258 ff.
  12. ↑ Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See in particular pp. 386 ff.
  13. ↑ Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See in particular p. 255.
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