We began with this opener (pdf). We then learned about inverse variation by investigation the relationship between distance, rate and time, d = rt (lesson, pdf).
Your homework for Monday is:
Figure out Tim Tebow's speed from this video (Tebow is at about the 2:41 mark). Bonus problem: Figure out Jacoby Ford's as well.
Do the problems on page 6 of the lesson (direct or inverse variation). You hopefully wrote them down in class, but, if not, get them from the lesson.
Do you need to re-assess over Solving Proportions and Percents? If so, prepare, make an appointment, and plan on coming in to re-assess.
Did you get behind on any of the homework this week? Or do you need to organize your notebook? This weekend would be a great time to get caught up on both of those.
Here's the deal: I'm working on curriculum for my school and Algebra 2 is making my eyes cross. I think the major problem is the state of Virginia is in a transition year between "old" Standards of Learning (SOLs), and "new" ones. This year is supposed to be the year that we're still teaching and assessing the old SOLs, but we're supposed to teach the new ones, too. Those of you that teach Algebra 2 already know that there's an enormous amount of information to cover in a short period of time. To give you context, our school teaches it as a semester-long block course. There's only so much a brain can handle in one day, though!
Here's the first draft of my skills list and structure...I'm not sure what to do about the old vs. new SOLs (my skills list is based on the old SOLs because that is what will be assessed).
Note: Gray items are not included in old or new SOLs but might be necessary for student understanding Blue items are being taken out of the SOLs starting next year Red items are new to the SOLs starting this year
Unit 1 Algebra 1 Review/Solving Equations
1 Solve multi-step equations and inequalities 2 Matrix +/- 3 Solve compound inequalities 4 Solve absolute value equations 5 Solve absolute value inequalities
Unit 2 Polynomial Review/Add Depth
6 Factor trinomial a = 1 7 Factor trinomial a > 1 8 Factor special cases (sum/diff of cubes, diff of squares, perfect square trinomials) 9 Factor out GCF first (factor completely) 10 Exponent rules 11 +/- polynomials 12 Multiply polynomials 13 Divide polynomials
Unit 3 Rational Expressions
14 Identify undefined values 15 Simplify rational expressions by factoring and canceling out common factors 16 Multiply and divide fractions 17 Multiply and divide rational expressions 18 Add and subtract fractions 19 Add and subtract rational expressions 20 Simplify complex fractions 21 Solve rational equations
Unit 4 Radicals, Radical Equations and Complex Numbers
22 Simplify numbers under radical 23 Simplify monomials under radical 24 Multiply and divide radicals 25 Add and subtract radicals 26 Nth roots to rational exponents and vice versa 27 Simplify expressions with nth roots and rational exponents 28 Solve radical equations 29 Simplify square roots with negative terms inside radical using i 30 Add and subtract complex numbers 31 Powers of i 32 Multiply complex numbers
Unit 5 Functions (intro)
33 Domain and range of relations (from ordered pairs, mapping, graph, table) 34 Identify relations that are functions and one-to-one 35 Given graph and a value k, find f(k) 36 Given graph, find zeros 37 Given graph and a value k, find where f(x)=k
Unit 6 Linear Functions
38 Slope from graph, equation, points 39 Graph from equation 40 Equation from graph 41 x- and y- intercepts 42 Determine whether lines are parallel, perpendicular, or neither from equation or graph 43 Write equations for parallel and perpendicular lines given line and point off the line 44 Graph linear inequalities
Unit 7 Systems
45 Solve systems of equations by graphing 46 Multiply Matrices using a graphing calculator 47 Inverse matrix method of systems 48 Systems of equations word problems 49 Graph systems of linear inequalities 50 Linear programming max/min problems
Unit 8 Functions (reprise)
51 Function math (addition, subtraction, multiplication, division) 52 Function composition, find a value i.e. f(g(3)) 53 Function composition, find the function i.e. f(g(x)) 54 Find an inverse function by switching variables
Unit 9 Quadratics
55 Graph from vertex form, identify max/min and zeros 56 Solve by factoring 57 Solve by Quadratic Formula (including complex solutions) 58 Determine roots using the discriminant 59 Write equation for quadratic given roots 60 Quadratic systems 61 Polynomials: relating x-intercept, zeroes and factors 62 End behavior for polynomials
Unit 10 Exponential/Logarithmic functions
63 Exponential growth or decay from function 64 Sketch base graph of exponential/log functions 65 Exponential to log and vice versa 66 Data analysis/curve of best fit for linear, quadratic, exponential and log
Unit 11 Transformations and Parent Functions
67 Graph absolute value functions 68 Horizontal and vertical translations of linear, quadratic, cubic, abs value, exponential and log 69 Reflections and stretching of linear, quadratic, cubic, abs value, exponential and log 70 Combinations of transformations on parent functions 71 Identify parent graphs of parent functions 72 Identify equations of parent functions
Unit 12 Conics
73 Identify a conic from graph 74 Identify a conic from equation
Unit 13 Variations
75 Write equation for direct, inverse and joint variation problems 76 Find the constant of variation
Unit 14 Sequences/Series
77 Write n terms of an arithmetic sequence 78 Find the sum of a finite arithmetic series 79 Write n terms of geometric sequence 80 Find sum of geometric series 81 Use formulas to find nth term 82 Identify sequence/series as arithmetic, geometric or neither
Unit 15 Statistics
83 Determine probabilities associated with areas under the normal crve 84 Compute permutations and combinations
If you made it this far, here's my call for help: Anyone have advice/suggestions for how to make this work and/or a better way to organize the information into cohesive units that seem to occur in a somewhat logical order? There is and will continue to be an emphasis on function families and transformations (as there should be). I find it difficult to express on paper how each function category needs to be a resting place, but they are all connected in the ways that transformations apply. Any ideas?
...oh...and I'm going to be teaching one section of deaf students and one section of blind students...in case that makes a difference
And here's the summary table for the linear model.
Call: lm(formula = V2 ~ V1)
Residuals: Min 1Q Median 3Q Max -1.7507 -0.7252 -0.1028 0.7953 2.2894
Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 60.54322 42.88214 1.412 0.169 V1 -0.03035 0.02150 -1.412 0.169
Residual standard error: 0.983 on 28 degrees of freedom Multiple R-squared: 0.06646, Adjusted R-squared: 0.03312 F-statistic: 1.993 on 1 and 28 DF, p-value: 0.1690
What does that model summary output mean? There really is NOT a significant downward trend of melt anomaly with years (there's no first order trend in fact). It's not measurably different from noise. How does this square with the recent reports of runaway melting though?
These recent posts about climate change stuff were inspired by a post I read about climate change skeptics, which I found because of my Google alerts on things related to 'computational fluid dynamics'. In the post she mentions Freeman Dyson, he's a pretty smart guy.
Concerning the climate models, I know enough of the details to be sure that they are unreliable. They are full of fudge factors that are fitted to the existing climate, so the models more or less agree with the observed data.
My favourite quote on the whole mess, a level-headed engineer from MIT quoted in a short article:
Mort Webster, assistant professor of engineering systems, warned that public discussion over climate change policies gets framed as a debate between the most extreme views on each side. "The world is ending tomorrow, versus it's all a myth," he said. "Neither of those is scientifically correct or socially useful."
An unusually intense period at work reduced my blogging activity to a few uninspired posts these past few weeks, but at long last I can return to the second part of my study on the application of labels such as “neighborhoods” and “subdivisions” to sub-districts within a larger metropolitan area. In the first part, I focused on the moneyed Garden District in Baton Rouge, which, in a city which is dominated by automobile-driven development patterns, emerges as one of the city’s most successful walkable, urban neighborhoods.
I use the term “neighborhood” loosely, primarily because, as I elaborated in Part I, the difference between a “neighborhood” and a &#! 8220;subdivision” often parallels the implied understanding of the distinctions between urban and suburban. Neighborhoods are old, urban, and walkable; subdivisions are newer, suburban, and auto-driven. These gross generalizations unfairly sequester the old and the new into two disparate categories, and the former enjoys a far loftier position in the cultural pecking order than the latter. Virtually everyone living in a reasonably dense residential community would like to claim part of a neighborhood, and civic associations rarely if ever organize themselves as the Highland Park Subdivision Association, for example. They use the word “neighborhood” instead. A realtor is far more likely to promote a home as “being part of a community with a genuine neighborhood feel”, and Mister Rogers immortalized his miniature Pittsburgh with the opening song “It’s a Beautiful Day in the Neighborhood”. These examples may seem facile, b! ut they demonstrate a prevailing aversion to “subdivisio! ns” ; as anything beyond a clinical term; it is the developer-speak that refers to the initial reorganization of land title through the accrued sale of individually subdivided plats, derived from an initially significantly larger parcel. But a good subdivision almost always strives to shed its tedious image, away from a series of financial agreements into something apparently much more organic—conceived from the aims and values of the people living there, rather than the paper-pushing of a businessperson with eyes on the dollar signs. In short, subdivisions always try to mature into neighborhoods.
This is precisely what has happened with the Garden District in Baton Rouge. As I noted before, an address in this neighborhood ranks among the most prestigious in the metro. But the Garden District didn’t begin with such a distinct identity. In fact, it consists of several smaller districts, which today are on ! the National Register, but began as subdivisions platted out by a private realty company.The area shaded in red is Roseland Terrace, platted in 1911; the region in blue is Drehr Place, a subdivision platted in 1921, while the green rectangle to the south of the other two historic districts is Kleinert Terrace, founded just a few years after Drehr Place. Though the three developments are contig! uous, they matured autonomously under these separate names for! decades . The Garden District Civic Association relates its origins on the website. Essentially a rezoning hearing in 1976 for a house on the northern edge of Drehr Avenue drew an unexpectedly large number of residents in the vicinity. After learning about their shared interest in its future and well-being, several members decided to form a neighborhood organization. They agreed to bestow upon it the name “Garden District” because of the positive connotations it arose, recalling the prestigious New Orleans neighborhood 80 miles downriver. Within a few weeks, some of these neighborhood activists (many of whom live in the area to this day) had drafted by-laws and elected a president. Thus, the Garden District as a neighborhood name and its respective neighborhood association were born simultaneously. To this day, the Civic Association ! collects dues, publishes newsletters, runs the adopt-a-tree program for live oaks in common spaces, and maintains the signs and bollards it installed to demarcate the neighborhood’s entrance. The original developers had conceived these three early subdivisions—Roseland Terrace, Drehr Place, and Kleinert Terrace—at slightly different time periods. B! ut as they aged comfortably, their residents witnessed new dev! elopment pushing considerably further to the north and east, and over time the architectural and socioeconomic similarities within the three early subdivisions became more widely visible. The eventual inception of a Civic Association was inevitable. It provided a forum for the transmission of the ideas and collective concerns that could reinforce the identity of a neighborhood. And through these regular meetings, the Association was able to bring to the table some of the technical specifications that the new Garden District was lacking—which, incidentally, happened to include the sort of “place-making” features that were increasingly prevalent in the subdivisions popping up in the outer suburbs. Essentially, the Garden District Civic Association—like so many others—has re-appropriated some of the initial roles of a developer. It clearly establishes cohesiveness to the neighborhood by simplifying the lines of communication. It also has implemented ! particular street and landscape improvements, which, in this day and age, would take place during the site planning stage, as a subdivision is getting off the ground.
By many standards, the neighborhood’s cohesiveness is patently visible (and no doubt was in 1976 as well). First, the array of housing types, while diverse, comes from a relatively uniform time period. The shared age of the housing mitigates the variety of styles and sizes.Secondly, the landscaping follows a certain basic pattern, emphasizing the tremendous tree canopy afforded by live oak trees, both in private property, and—in the wider streets—spaces throughout the broad, grassy medians. If the front yards cannot fit such an expansive tree, they will often host smaller indigenous species, such as the crepe myrtle.Thirdly, the three subdivisions share borders, proven by the three colored transparencies on the oft-referenced map.
Fourth, and perhaps the most important for the arguments featured in this half of the blog post, the three smaller subdivisions that make up the Garden District all sh! are the same gridded street network. This shared network affords them a high level of interpenetrability among all the streets that make up the three, as well as—and this is critical—the surrounding neighborhoods.The strange brown shape shows the means of accessing the Roseland Terrace and Drehr Place involves nothing more than crossing an intersection. Even more critical is the! purple demarcation on the map below:That thick purple line essentially separates the affluent sections of this part of Baton Rouge from the poor ones. I hate to make such broad distinctions on something as simple as a street map, but empirical evidence generally supports this. The Garden District occupies the northwestern most portion of a mostly upper-middle class part of town, but directly to the north and west of the neighborhood’s boundaries are considerably po! orer districts. Particularly noticeable is the neighborhood west of 18th Street, where the housing more frequently looks like this:Simple, unadorned single shotgun houses with virtually no front yard and little foliage.Unlike in the Garden District, where cables are either buried or hidden behind back alleys, here they are out front along the streets and sidewalks. And the old commercial buildings along Government Street (the northern boundary to the Garden District) are generally vacant in the area to the west of the affluent neighborhood. Here is the vista at around 16th and Government:As Government Street continues toward the wealthier Garden District, the retail landscape is hardly top-tier for the metro, but it is considerably stronger than the virtual abandonment visible in the picture above.
In short, the Garden District—a general term for three old-money subdivisions in inner-city Baton Rouge—sits cheek-by-jowl with one of the poorer old neighborhoods, with not even a major arterial street or the stereotypical railroad track to separate them. For example, at the corner of 18th and Cherokee, on the edge of Roseland Terrace, one sees a typical! ly immaculate house characteristic of the Garden District.And just a two blocks to the west, on 16th Street, the view below is not uncommon:In all likelihood, the neighborhood to the west of 18th street was not impoverished at the time Roseland Terrace was platted on the site of a former fairground racetrack. It was probably a working class or lower-middle class community. But by the time the Civic Association organized itself and bestowed the name Garden District to these three subdivisions, the fortunes on the two sides of 18th Street had diverged significantly. Forty years later, Old South Baton Rouge remains a largely low-income African American community, while the Garden District is mostly white and virtually devoid of poverty. In an era in which discriminatory redlining, fraudulent blockbusting, and publicly sanctioned segregation (onc! e common in the South) are all illegal, how can two neighborhoods show such significant disparities, with the desirability of the Garden District remaining superlative despite sitting so close to such poverty? The Garden District Civic Association undoubtedly provides many of the answers.Aside from organizing a garden club to protect the live oaks, social committees to plan Christmas caroling and ice cream g! atherings, or public relations to organize home tours, the Civ! ic Assoc iation also has hired a separate security unit on top of the existing Baton Rouge Police Department. (Incidentally, the original Garden District of New Orleans, which sits almost as close to an even more impoverished neighborhood, also hires a plainly visible separate security force.) The association also allows homeowners who will be out of town for a lengthy amount of time to report their unoccupied home for extra monitoring during their absence.
Such actions are hardly unique among urban neighborhood associations, particularly those that are wealthy but remain close to considerably poorer areas. I by no means am attempting to portray the Garden District Civic Association nor the neighborhood’s residents as exclusive or prejudiced. They are reacting in a similar fashion as many other wealthy districts that rest squarely in high-crime cities, and the residents have clearly opted to buy into unity of activit! ies as well as a shared sense of added security by remaining in the neighborhood instead of abandoning it to newer subdivisions out in the suburbs, where they would undoubtedly be far removed from inner-city privations and violence.
What is interesting about this is that, perhaps more powerfully than just signage and tree plantings, the Civic Association is helping to foster the unity an urban neighborhood needs, quite possibly as a compensatory gesture for the fact that its street configuration cannot exclude strangers in any other way. Compare the street grid from the color-coded maps above to the one below, several miles away on Highland Road, one of the wealthiest suburban districts within Baton Rouge city limits:Most of the housing around here post-dates the 1960s. Street designers for these subdivisions/neighborhoods have all but abandoned the old grid for a hierarchical design, in which most of the streets terminate in cul-de-sacs. Each individual development usually has only one or two means of ingress, as opposed to the Garden District, which has closer to twenty. The development pattern becomes even more pronounced a few miles further out on Highland Road.Here, in the last outskirts of the city and East Baton Rouge Parish, the housing typically post-dates the 1980s and is almost uniformly wealthy. The subdivisions are smaller and even less interconnected. Some of them are gated at the front.
Any elementary student of urban studies has caught on to this long ago; the average layperson can also recognize a change in street configuration from the old inner-city neighborhood and the modern subdivision. What is most striking is how urban neighborhoods have essentially had to co-opt certain features from suburban subdivisions—as well as! duplicate basic city services—in order to preserve thei! r desira bility. The Garden District in Baton Rouge cannot build gates around every one of its entrances to keep the higher criminal activity at bay that residents might associate with the neighborhood to the west. Instead, the Civic Association must find ways to cultivate unity and inclusivity from within. It uses carrots like the neighborhood picnics, organized garage sales, decorative signage and lighting, and recommended arborists, electricians, or carpenters. It depends upon sticks as well, such as the additional security, stipulations for people who rent out part or all of their property, and a clear line of communication for reporting of city code violations, such as parking in the grass or sweeping debris into the street. Neighborhoods such as the Garden District will depend on a certain capacity to exclude on paper to compensate for the inability to exclude via physical barriers. One may consider this elitist or racist, but neighborhood associations are so prevalent in! this day and age that they hardly single out certain segments of society—persons of all strata may find their neighborhood has one, rich or poor, white or black, urban or suburban.
The superficial stereotypes I listed in Part I of this blog post clearly placed neighborhoods into one categorical box and subdivisions into another. I did this ironically, because the old, venerable Garden District neighborhood originated from the subdivision of real estate that followed a similar pattern to the exurban subdivisions of today. Subdivisions and neighborhoods are little more than expressions of individual preferences; the line distinguishing them is impossibly blurry. The biggest difference, of course, is that the old urban subdivisions (which we are more likely to refer to as “neighborhoods”) typically depended upon rectilinear grids with an almost unlimited number of points of entry. Conversely, most! contemporary neighborhoods (often referred to as “subdi! visions& #8221; until they mature and develop something we like to call “character”) employ curvilinear cul-de-sacs that allow for a much clearer monitoring of who comes and goes—which nearly always happens by vehicle instead of by foot. One could argue, as many urbanists do, that the grid is more desirable because it offers a greater freedom of mobility. But the non-gridded subdivision emerged as a reaction to the characteristics that people found least appealing about the historic grid: namely, the ability to restrict who enters, whether it involves speeding cars that cut through or potential ne’er-do-wells from a neighboring community. The Garden District in Baton Rouge has undoubtedly attracted a certain type of resident that consciously eschews the suburban cul-de-sac, but such individuals often find themselves devoting more time and money to retain a certain level of security and privacy that almost everyone hopes to attain. The Civic Association has! found a generally benign way of achieving this, keeping a gridded neighborhood safe and attractive as the majority of the population continues to surge toward cul-de-sacs ten miles down the road. powerful last names
We have seen in the last two postings that a Fourier series is intimately linked to sines and cosines. In the last posting we also discovered that there are two forms of Fourier series using sines and cosines. One form uses cosines with phases and one form uses sines and cosines without phases. In this posting we will explore this concept in more depth.
Addition of functions
Animation illustrating the step-by-step process of adding two functions. This shows the laborious process of adding the functions together at each x-position. Mouse over the animation to start it, off to suspend it, click to restart. You can also use the buttons at the top. You might explore the various functions. Some may serve to illustrate this idea better for you.
Fourier series involves addition of functions. Mathematically we might express such an addition as:
. (1)
This means that for each x, we add the first function's y value at this x to the second function's y value at the same x. This gives us the sum function's y value for this x.
While this is easy to express as an equation, it is rather exhausting to actually do in practice. The animation at the right illustrates this for a number of functions. Mouse and click on the animation to start it. Mouse over the fast button to speed it up, or click on the step button to have it go a step at a time. Read the caption at the bottom for a step-by-step explanation. Clicking "change f1" or "change f2" will change the functions used. Clicking on the animation will restart it.
The animation shows three functions: a red one on the top graph, a blue one on the middle graph, and a purple one on the lower graph. The animation shows the process of going point-by-point to sum the upper two functions to make the lower, sum function.
The first set of functions in the animation consists of two sinusoids of the same wavelength. A little observing of the animation will show you that the addition of two same wavelength sinusoids results in another sinusoid of the same wavelength. This concept is a very important point in this posting.
Relationship between sine and cosine functions
As was briefly explained in the last posting, the sine and the cosine functions are very similar. The two are shown in the graph at the right. The sine function is identical to the cosine function except that the sine function is shifted to the right by 90°. Another way to achieve a 90° right shift is by subtracting 90° from the argument of the cosine function, i.e. using cos(θ − 90°) in place of cosθ (you can use this method to shift any function over... subtract a constant from its argument). We would call this a −90° phase shift. Or alternately, we can say that the cosine function is a sine function shifted 90°, i.e. to the left. There are many references on the internet for the relationship between sines and cosines (google on "trig identities"). The equations for the relationships used here are:
sinθ = cos(θ − 90°)
cosθ = sin(θ + 90°)
(2)
As was mentioned in the last posting, we can add mixtures of sine and cosine functions together to produce a cosine with various amplitudes and phases. The animation below illustrates this. The red and blue functions are cosine and sine functions, respectively, each multiplied by an expanding or reducing coefficient. The sum of the two functions is shown in purple. The purple function can be arrived at by adding the two functions together, x position by x position, or by using phasors, as is shown on the right side of the animation. The viewer can drag the squares of the sine and cosine functions to experiment with changing the amplitudes of these functions. The viewer can also drag the amplitude and phase of the sum function and see what sine and cosine amplitudes are required to produce various combinations of amplitude and phase.
The backgrounds of the graphs are colored red and blue to indicate the regions of the sum function that are dominated by either the red cosine function or the blue sine function. The phasor dial is a graphical way to represent the phase of the functions, similar to the way the phase of the moon is represented on some clocks. The use of phasors to add sinusoidal functions is explained (and animated) in the earlier posting titled "phasors". Look at the paragraph titled "Adding two waves in a phasor diagram".
Animation showing how sine and cosine functions with adjustable amplitudes can add up to form a cosine function with any amplitude and phase. Both the functions and their phasors are shown. The user can drag the handles (small colored squares) to adjust the amplitudes of the sine and cosine functions and see the result of their summation. Or the user can adjust the amplitude and/or phase of the sum function and see the changes in amplitudes of the sine and cosine functions necessary to produce this sum function. Alternately, the user can adjust the phasors to accomplish the same results. You can also type in different values in the equations (hit "enter" after making the change.)
You might try dragging the purple sum function handle in the left graph horizontally and observe the movement of the phasor. Or, observe how the pure sine and cosine functions rise and fall as needed to accommodate the changing phase.
The mathematical relationships between the amplitudes of the sine and cosine functions and the amplitude and phase of their sum are:
(3)
(4)
(5a) (5b)
(6)
(7)
The first equation simply states that the cosine function with amplitude A3 and phase φ is the sum of cosine and sine functions with respective amplitudes A1 and A2. In a sense it defines the constants A1, A2, A3, andφ. We use the independent variable θ here but other variables could easily be substituted for θ, such as x. The second line of equations gives the amplitude and phase of the sum in terms of the amplitudes of the cosine and sine functions. Note that the phase φ is best given in terms of the arctan2 function, as seen in many programming languages, which returns angles over the entire 360° or 2π radians. The third line of equations gives the inverse of the equations in the second line, the amplitudes of the sine and cosine functions required so that they sum to a cosine function with amplitude A3 and phase φ. These relationships are derived in the box below at the right.
Derivation of the relationships in the previous table
We start with the standard relation for the cosine of the sum of two angles: Applying this to the left side of equation (3) in the previous table, we have: Comparing this with the right side of equation (3), we see that the first term in parenthesis is the coefficient of cosθ or A1, i.e. A1 = A3 cos φ which is equation (6) above.
The second term in parenthesis (the coefficient of sinθ) is, according to equation (3), A2 so that A2 = −A3 sin φ which is equation (7).
Now we use our just proven equations (6) and (7). To get equation (4), we simply square both sides of equations (6) and (7) and add them together: , or where in the last step we have used the fact that the sine squared (of any variable) plus the cosine squared (of that same variable) equals one. We solve the last expression for A3 to give us equation (4).
We get equation (5) by dividing equation (7) by equation (6) and canceling out the A3's that are in both the numerator and the denominator of the right side. We then apply the inverse (i.e. arc) tangent of both sides.
Relationship of the above formulas to the two forms of Fourier series
In the last posting we presented the following two versions of Fourier series:
(8),
(9)
These two versions are related to each other via the equations (3) through (7) above. Of course we need to change the variable names of the amplitudes and also replace θ with nθ. Equation (3) shows the overall relationship that allows us to change equation (8) into equation (9), whereas equations (4) through (7) show the relationship between the amplitudes and phases in equation (8) and equation (9). Changing the names of the amplitudes as appropriate and replacing θ with nθ, we have:
(10)
(11)
(12a) (12b)
(13)
(14)
As we said in the previous posting, version (8) of the Fourier series equation is easier to understand, while version (9) is a much easier form to compute the coefficients for, if we are given a function. Equations (11) through (14) allow us to convert one version into the other.
Review of the example done in the last posting:
At the end of the previous posting we computed the coefficients for version (9) using the standard equations at the left below on the waveform shown in the center below:
Equations, waveform, and results of the previous posting's example.
(15)
(16)
(17)
We obtained the result that the constant term a0 was ½. All the cosine terms were zero, i.e. all the an's equaled zero. Also half the bn's were zero as well (those with even n numbers). The odd sine terms had coefficients given by: bn = 2/nπ and are graphed in the right-most panel above.
Using equation (11) with these a's and b's, we see that the amplitudes, the An's, are given as follows: for even n's, An = 0 while for odd n's An = 2/nπ. Using equation (12), we see that the φn's for even n's are indeterminate (they equal 0/0), while for odd n's the phase shifts are given by φn = −π/2 .
The An's are the amplitudes of the various Fourier components while the φn's are the phase shifts. For this particular function, a graph of the An's versus n would look exactly like the right panel above, except that the vertical axis would be labeled An and not bn.
Repeating the exercise with a shifting of the function:
For another example, consider shifting the waveform above, so that now it is symmetrically placed on either side of the y axis, as show at the left below. We use integration from 0 to 2π as we did in the previous example, but we could equally well have integrated from −π to +π and gotten the same results. This might be a good exercise for a student to try. The rule is that you need to integrate over exactly one cycle of the repeating waveform.
Example 2. Symmetrical square wave.
Function to be broken into Fourier components
Calculation of the constant term
Calculation of the cosine coefficients
Calculation of the sine coefficients
or
The cosine coefficients for example 2. Note that they alternate in sign.
Looking at the above results, we see that the constant term is still one half. This is as we would expect, since the average value will remain unchanged during a simple shifting to the right or left. However the other coefficients, the an's and bn's, do change. In our new case, the sine coefficients, the bn's, are all zero and the cosine coefficients, the an's, are the same as the sine coefficients of the previous example, except that they now alternate in sign. Shifting over has interchanged the values of these two sets of coefficients and added the ± sign. Someone experienced with Fourier series would say a lot of this is obvious, that after subtracting the constant term, the first example is an anti-symmetric function ( f(−θ) = −f(θ) ), while the second example is a symmetric function ( f(−θ) = f(θ) ). To an experienced person, this would mean that the sinusoid components in the first case would be anti-symmetric, i.e. sine functions, while those in the second case would be symmetric, i.e. cosine functions.
If we use equations (11) and (12), we find that the An's are just as before: An = 0 for all even n's and An = 2/nπ for all odd n's. On the other hand, the phases have changed: while the φn's for even n's are still indeterminate, the φn 's for odd n's alternate between 0 and π (i.e. φn = 0 and π). In the previous example φn = −π/2. Thus, the phases have changed.
As a general rule, shifting a waveform in the horizontal direction does not change the An's but it does change the relative sizes of the a's and b's and the phases, the φn's. Although we haven't demonstrated this here, it is also true that shifting the function vertically only changes the constant term a0 and not the other coefficients, the an's and bn's. An interesting exercise for students is to try this, i.e. add the constant 2 to the above function and redo the calculations.
The following table summarizes the comparison of examples 1 and 2.
Comparison of examples 1 and 2
Example 1 square wave↓
Example 2 - symmetric square wave - shifted↓
Comparison↓
Function to be analyzed →
Example 2 is shifted to the left, i.e. by a phase of + radians.
Spectrum →
The two spectra are the same.
Phases →
Concerning the labeling of some phases as "indeterminate" (n's for which An = 0, i.e. zero amplitude). Equations (13) and (14) above indicate that zero amplitude (An = 0) means that an = bn = 0. Substitution into equation (12) yields an indeterminate φn. In terms of phasors, this is equivalent to saying that a phasor (or vector) with zero length has an indeterminate direction.
n
φn
1
−π/2
2
indeterminate
3
−π/2
4
indeterminate
5
−π/2
n
φn
1
0
2
indeterminate
3
π
4
indeterminate
5
0
The phases of the non-zero components of example 2 are shifted by + and − radians.
Conclusion: shifting the function in the horizontal direction changes the phases of the various Fourier components but does not change the amplitudes of these components.
Click and drag the small turquoise square next to the top of the frame to move the square wave function and see the effect that the shift has on the function's Fourier series. Note that while the an's, bn's, and φn's change, the An's remain unaffected by shifting the function (except for A0 that does change with vertical shifts). Note also that as you move the function horizontally, the purple phasor vectors rotate. The vectors having higher n's rotate the fastest.
At the right, we include an animation that allows the user to drag a square wave function horizontally and vertically and see the various coefficients change as a function of the dragging. The square wave has an amplitude of one. To translate the results into a larger amplitude square wave, the user needs to multiply all coefficients by the larger amplitude. The phase (shown in degrees) would not change. The table of numbers at the very bottom show the numerical values of the coefficients, while the bar charts and vectors (phasors) are graphical representations. Note that positive sine coefficients, i.e. the b's, are represented by downward pointing arrows in the phasors diagrams. The user can drag the square wave function around and verify the results discussed in previous paragraphs.
In the table following the animation, we derive the math used in the animation. The math is based on the function shown in the first cell of the table. The function has a horizontal shift from example 1 of δ and a vertical shift of y0 . The other cells in the table show the computation of the various coefficients.
Example 3. General shifted square wave.
Function to be broken into Fourier components
Calculation of the constant term
In the second line we have used the result of calculating a0 in example 1 of the previous posting, i.e. the constant term (or average value) of a square wave is one half its amplitude. The final result is rather obvious, that the average value of a square wave equals the vertical offset of its base plus one half its height.
Calculation of the cosine coefficients
Calculation of the sine coefficients
Thus the a's are zero for all even n's and equal to −2sinnδ/nπ for odd n's.
In example 1, δ = 0, so our equation just above becomes an = 0 for all n's. This agrees with the earlier result in example 1. In example 2, δ = −π/2. Using the above equation, we get an = ±2/nπ for odd n's and 0 for even n's, after some careful substituting of various n's. This agrees with the above results.
So similar to the a's, the b's also are zero for even n's. In the case of odd n's, the b's are given by 2cosnδ/nπ.
Calculation of spectral amplitudes
Calculation of spectral phases
Since the an's and bn's are zero for even n's, the An's are also zero for even n's. For odd n's, using the equations above, we get: This is in agreement with examples 1 and 2 above. It confirms that the spectral amplitudes are not affected by a function's position or offset.
This is asking what vector with angle φ with respect to the x axis would result in coordinates (at the vector tip) of −bn in the y direction and an in the x direction. Since the an's and bn's are zero for even n's and there is no information about vector direction, φ is indeterminate for even n's. For odd n's, we substitute the above values to get:
So what vector direction would result in a vector tip being at x = −M sin nδ and y = −M cos nδ? The magnitude M is given by M = 2/nπ. The adjacent sketch shows the orientation required so that the vector tip has these x and y coordinates. Since positive φ is measured counterclockwise from the x axis (and negative φ is clockwise), we see that the vector in the sketch has a φ given by: Example 1 had δ = 0 and phases of −π/2 for odd n's. This agrees with our above equation. Example 2 had δ = −π/2. Using the above equation for odd n's, we get:
which agrees with our earlier result for example 2.
