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| {{refimprove|date=June 2009|bot=yes}}
| | The initial website I would like to share with you is www.startyourdiet.com. We have to register for this site yet a simple membership is completely free. This website has many tools to aid we shed the pounds. There are tools for daily fat monitoring, a tool to program weight reduction goals, an automated goal planning tool, a diet profile, an advanced meal planner and tracker, a daily food journal, along with a weight-tracking chart. There is equally a calculator for we to figure the body mass index (BMI). www.startyourdiet.com also offers an online community for help as you lose fat. We additionally receive your own blog.<br><br>There is not any unique event or dramatic occurrence when we cross the line above the BMI of 25 or 30. It is just a reference point to which the improved risk of obtaining weight-related condition happens whenever you are inside those ranges.<br><br>Choose low-fat foods, watch how several calories you may be taking in per meal, and eat a balanced meal. Balanced meals could have a lot of vegetables, fruits, entire grains, low-fat dairy, and lean meats and proteins. Eat because little fat as potentially inside your diet.<br><br>You can find the answer for 'am I overweight for my age and height' on your. All you want is calculate the BMI and see when the value is falling in the suggested range or not. Remember which the formula for calculating BMI differs for kids and adults. Thus, to locate out BMI for kids, use the correct BMI calculating formula for kids. Likewise, folks that are older than 20 years could [http://safedietplans.com/bmi-calculator bmi calculator women] for adults.<br><br>But perhaps the researches have noticed something important. In the past, big was beautiful. Overweight people were considered more attractive, more affluent, and healthier. Since the advent of the camera, which supposedly "adds ten pounds," skinny individuals have become the hot idols to emulate. In Quentin Tarantino's Pulp Fiction, a woman says, "it's unfortunate what you find pleasing to the touch plus pleasing to the eye is rarely the same." Is that regarding to change?<br><br>The simplest method to keep track of your calorie and fat intake for the day is to keep a log! Hide a small notepad and pen in the purse especially for this and take it wherever we go. Write down what we eat for the day. A perfect, free website to calculate how countless calories are inside your food is a piece of iGoogle!<br><br>If BMI is not the appropriate tool, what exactly is? Well, to certainly precisely tell what percent of your weight is fat, you need to be weighed underwater. That's a hassle, not the sort of thing the average person will do, certainly not on a regular basis. But, with just a tape measure, you are able to calculate the waist-hip ratio. Next you can utilize http://www.healthcalculators.org/calculators/waist_hip.asp to find if it's in a healthy range. This really is a much more sensible measure. |
| The '''bilinear transform''' (also known as '''[[Arnold Tustin|Tustin]]'s method''') is used in [[digital signal processing]] and discrete-time [[control theory]] to transform continuous-time system representations to discrete-time and vice versa. | |
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| The bilinear transform is a special case of a [[conformal map]]ping (namely, the [[Möbius transformation]]), often used to convert a [[transfer function]] <math> H_a(s) \ </math> of a [[linear]], [[time-invariant]] ([[LTI system theory|LTI]]) filter in the [[continuous function|continuous]]-time domain (often called an [[analog filter]]) to a transfer function <math> H_d(z) \ </math> of a linear, shift-invariant filter in the [[discrete signal|discrete]]-time domain (often called a [[digital filter]] although there are analog filters constructed with [[switched capacitor]]s that are discrete-time filters). It maps positions on the <math> j \omega \ </math> axis, <math> Re[s]=0 \ </math>, in the [[s-plane]] to the [[unit circle]], <math> |z| = 1 \ </math>, in the [[complex plane|z-plane]]. Other bilinear transforms can be used to warp the [[frequency response]] of any discrete-time linear system (for example to approximate the non-linear frequency resolution of the human auditory system) and are implementable in the discrete domain by replacing a system's unit delays <math> \left( z^{-1} \right) \ </math> with first order [[all-pass filter]]s.
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| The transform preserves [[BIBO stability|stability]] and maps every point of the [[frequency response]] of the continuous-time filter, <math> H_a(j \omega_a) \ </math> to a corresponding point in the frequency response of the discrete-time filter, <math> H_d(e^{j \omega_d T}) \ </math> although to a somewhat different frequency, as shown in the [[#Frequency warping|Frequency warping]] section below. This means that for every feature that one sees in the frequency response of the analog filter, there is a corresponding feature, with identical gain and phase shift, in the frequency response of the digital filter but, perhaps, at a somewhat different frequency. This is barely noticeable at low frequencies but is quite evident at frequencies close to the [[Nyquist frequency]].
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| == Discrete-time approximation ==
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| The bilinear transform is a first-order approximation of the natural logarithm function that is an exact mapping of the z-plane to the s-plane. When the [[Laplace transform]] is performed on a discrete-time signal (with each element of the discrete-time sequence attached to a correspondingly delayed [[Dirac delta function|unit impulse]]), the result is precisely the [[Z transform]] of the discrete-time sequence with the substitution of
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| :<math>
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| \begin{align}
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| z &= e^{sT} \\
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| &= \frac{e^{sT/2}}{e^{-sT/2}} \\
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| &\approx \frac{1 + s T / 2}{1 - s T / 2}
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| \end{align}
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| </math>
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| where <math> T \ </math> is the [[numerical integration]] step size of the [[trapezoidal rule]] used in the bilinear transform derivation.<ref>{{cite book |title=Discrete Time Signal Processing Third Edition |last=Oppenheim |first=Alan |year=2010 |publisher=Pearson Higher Education, Inc. |location=Upper Saddle River, NJ |isbn=978-0-13-198842-2 |page=504}}</ref> The above bilinear approximation can be solved for <math> s \ </math> or a similar approximation for <math> s = (1/T) \ln(z) \ \ </math> can be performed.
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| The inverse of this mapping (and its first-order bilinear approximation) is
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| :<math>
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| \begin{align}
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| s &= \frac{1}{T} \ln(z) \\
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| &= \frac{2}{T} \left[\frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + \frac{1}{5} \left( \frac{z-1}{z+1} \right)^5 + \frac{1}{7} \left( \frac{z-1}{z+1} \right)^7 + \cdots \right] \\
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| &\approx \frac{2}{T} \frac{z - 1}{z + 1} \\
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| &= \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}}
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| \end{align}
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| </math>
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| The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function, <math> H_a(s) \ </math>
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| :<math>s \leftarrow \frac{2}{T} \frac{z - 1}{z + 1}.</math>
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| That is
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| :<math>H_d(z) = H_a(s) \bigg|_{s = \frac{2}{T} \frac{z - 1}{z + 1}}= H_a \left( \frac{2}{T} \frac{z-1}{z+1} \right). \ </math>
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| == Stability and minimum-phase property preserved ==
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| A continuous-time causal filter is [[BIBO stability|stable]] if the [[Pole (complex analysis)|poles]] of its transfer function fall in the left half of the [[complex number|complex]] [[s-plane]]. A discrete-time causal filter is stable if the poles of its transfer function fall inside the [[unit circle]] in the [[complex plane|complex z-plane]]. The bilinear transform maps the left half of the complex s-plane to the interior of the unit circle in the z-plane. Thus filters designed in the continuous-time domain that are stable are converted to filters in the discrete-time domain that preserve that stability.
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| Likewise, a continuous-time filter is [[minimum-phase]] if the [[Zero (complex analysis)|zeros]] of its transfer function fall in the left half of the complex s-plane. A discrete-time filter is minimum-phase if the zeros of its transfer function fall inside the unit circle in the complex z-plane. Then the same mapping property assures that continuous-time filters that are minimum-phase are converted to discrete-time filters that preserve that property of being minimum-phase. | |
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| == Example ==
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| As an example take a simple [[low-pass]] [[RC filter]]. This continuous-time filter has a transfer function
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| :<math>\begin{align} | |
| H_a(s) &= \frac{1/sC}{R+1/sC} \\
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| &= \frac{1}{1 + RC s}.
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| \end{align}</math>
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| If we wish to implement this filter as a digital filter, we can apply the bilinear transform by substituting for <math>s</math> the formula above; after some reworking, we get the following filter representation:
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| :{|
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| |-
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| |<math>H_d(z) \ </math>
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| |<math> =H_a \left( \frac{2}{T} \frac{z-1}{z+1}\right) \ </math>
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| |-
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| |<math>= \frac{1}{1 + RC \left( \frac{2}{T} \frac{z-1}{z+1}\right)} \ </math>
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| |-
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| |<math>= \frac{1 + z}{(1 - 2 RC / T) + (1 + 2RC / T) z} \ </math>
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| |-
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| |<math>= \frac{1 + z^{-1}}{(1 + 2RC / T) + (1 - 2RC / T) z^{-1}}. \ </math>
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| |}
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| The coefficients of the denominator are the 'feed-backward' coefficients and the coefficients of the numerator are the 'feed-forward' coefficients used to implement a real-time [[digital filter]].
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| == Frequency warping ==
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| To determine the frequency response of a continuous-time filter, the [[transfer function]] <math> H_a(s) \ </math> is evaluated at <math>s = j \omega \ </math> which is on the <math> j \omega \ </math> axis. Likewise, to determine the frequency response of a discrete-time filter, the transfer function <math> H_d(z) \ </math> is evaluated at <span style="vertical-align:+30%;"><math>z = e^{ j \omega T} \ </math></span> which is on the unit circle, <math> |z| = 1 \ </math>. When the actual frequency of <math> \omega \ </math> is input to the discrete-time filter designed by use of the bilinear transform, it is desired to know at what frequency, <math> \omega_a \ </math>, for the continuous-time filter that this <math> \omega \ </math> is mapped to.
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| :<math>H_d(z) = H_a \left( \frac{2}{T} \frac{z-1}{z+1}\right) \ </math>
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| :{|
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| |-
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| |<math>H_d(e^{ j \omega T}) \ </math>
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| |<math>= H_a \left( \frac{2}{T} \frac{e^{ j \omega T} - 1}{e^{ j \omega T} + 1}\right) \ </math>
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| |-
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| |<math>= H_a \left( \frac{2}{T} \cdot \frac{e^{j \omega T/2} \left(e^{j \omega T/2} - e^{-j \omega T/2}\right)}{e^{j \omega T/2} \left(e^{j \omega T/2} + e^{-j \omega T/2 }\right)}\right) \ </math>
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| |-
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| |<math>= H_a \left( \frac{2}{T} \cdot \frac{\left(e^{j \omega T/2} - e^{-j \omega T/2}\right)}{\left(e^{j \omega T/2} + e^{-j \omega T/2 }\right)}\right) \ </math>
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| |-
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| |<math>= H_a \left(j \frac{2}{T} \cdot \frac{ \left(e^{j \omega T/2} - e^{-j \omega T/2}\right) /(2j)}{\left(e^{j \omega T/2} + e^{-j \omega T/2 }\right) / 2}\right) \ </math>
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| |-
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| |<math>= H_a \left(j \frac{2}{T} \cdot \frac{ \sin(\omega T/2) }{ \cos(\omega T/2) }\right) \ </math>
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| |-
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| |<math>= H_a \left(j \frac{2}{T} \cdot \tan \left( \omega T/2 \right) \right) \ </math>
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| |}
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| This shows that every point on the unit circle in the discrete-time filter z-plane, <span style="vertical-align:+30%;"><math>z = e^{ j \omega T} \ </math></span> is mapped to a point on the <math>j \omega \ </math> axis on the continuous-time filter s-plane, <math>s = j \omega_a \ </math>. That is, the discrete-time to continuous-time frequency mapping of the bilinear transform is
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| :<math> \omega_a = \frac{2}{T} \tan \left( \omega \frac{T}{2} \right) </math>
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| and the inverse mapping is
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| :<math> \omega = \frac{2}{T} \arctan \left( \omega_a \frac{T}{2} \right). </math>
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| The discrete-time filter behaves at frequency <math>\omega \ </math> the same way that the continuous-time filter behaves at frequency <math> (2/T) \tan(\omega T/2) \ </math>. Specifically, the gain and phase shift that the discrete-time filter has at frequency <math>\omega \ </math> is the same gain and phase shift that the continuous-time filter has at frequency <math> (2/T) \tan(\omega T/2) \ </math>. This means that every feature, every "bump" that is visible in the frequency response of the continuous-time filter is also visible in the discrete-time filter, but at a different frequency. For low frequencies (that is, when <math>\omega \ll 2/T</math> or <math>\omega_a \ll 2/T</math>), <math>\omega \approx \omega_a \ </math>.
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| One can see that the entire continuous frequency range
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| : <math> -\infty < \omega_a < +\infty \ </math>
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| is mapped onto the fundamental frequency interval
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| : <math> -\frac{\pi}{T} < \omega < +\frac{\pi}{T}. \ </math>
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| The continuous-time filter frequency <math> \omega_a = 0 \ </math> corresponds to the discrete-time filter frequency <math> \omega = 0 \ </math> and the continuous-time filter frequency <math> \omega_a = \pm \infty \ </math> correspond to the discrete-time filter frequency <math> \omega = \pm \pi / T. \ </math> | |
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| One can also see that there is a nonlinear relationship between <math> \omega_a \ </math> and <math> \omega. \ </math> This effect of the bilinear transform is called '''''frequency warping'''''. The continuous-time filter can be designed to compensate for this frequency warping by setting <math> \omega_a = \frac{2}{T} \tan \left( \omega \frac{T}{2} \right) \ </math> for every frequency specification that the designer has control over (such as corner frequency or center frequency). This is called '''''pre-warping''''' the filter design.
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| When designing a digital filter as an approximation of a continuous time filter, the frequency response (both amplitude and phase) of the digital filter can be made to match the frequency response of the continuous filter at frequency <math> \omega_0 </math> if the following transform is substituted into the continuous filter transfer function.<ref>Astrom, Karl J. ''Computer Controlled Systems, Theory and Design'' Second Edition. ISBN 0131686003. Prentice-Hall, 1990, pp 212</ref> This is a modified version of Tustin's transform shown above. However, note that this transform becomes the above transform as <math> \omega_0 \to 0 </math>. That is to say, the above transform causes the digital filter response to match the analog filter response at DC.
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| :<math>s \leftarrow \frac{\omega_0}{\tan(\frac{\omega_0 T}{2})} \frac{z - 1}{z + 1}.</math>
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| The main advantage of the warping phenomenon is the absence of aliasing distortion of the frequency response characteristic, such as observed with [[Impulse invariance]]. It is necessary, however, to compensate for the frequency warping by pre-warping the given frequency specifications of the continuous-time system. These pre-warped specifications may then be used in the bilinear transform to obtain the desired discrete-time system.
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| ==See also==
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| * [[Impulse invariance]]
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| * [[Matched Z-transform method]]
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| ==References==
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| {{refimprove|date=February 2011}}
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| {{reflist}}
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| {{DSP}}
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| {{DEFAULTSORT:Bilinear Transform}}
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| [[Category:Digital signal processing]]
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| [[Category:Transforms]]
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| [[Category:Control theory]]
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There is not any unique event or dramatic occurrence when we cross the line above the BMI of 25 or 30. It is just a reference point to which the improved risk of obtaining weight-related condition happens whenever you are inside those ranges.
Choose low-fat foods, watch how several calories you may be taking in per meal, and eat a balanced meal. Balanced meals could have a lot of vegetables, fruits, entire grains, low-fat dairy, and lean meats and proteins. Eat because little fat as potentially inside your diet.
You can find the answer for 'am I overweight for my age and height' on your. All you want is calculate the BMI and see when the value is falling in the suggested range or not. Remember which the formula for calculating BMI differs for kids and adults. Thus, to locate out BMI for kids, use the correct BMI calculating formula for kids. Likewise, folks that are older than 20 years could bmi calculator women for adults.
But perhaps the researches have noticed something important. In the past, big was beautiful. Overweight people were considered more attractive, more affluent, and healthier. Since the advent of the camera, which supposedly "adds ten pounds," skinny individuals have become the hot idols to emulate. In Quentin Tarantino's Pulp Fiction, a woman says, "it's unfortunate what you find pleasing to the touch plus pleasing to the eye is rarely the same." Is that regarding to change?
The simplest method to keep track of your calorie and fat intake for the day is to keep a log! Hide a small notepad and pen in the purse especially for this and take it wherever we go. Write down what we eat for the day. A perfect, free website to calculate how countless calories are inside your food is a piece of iGoogle!
If BMI is not the appropriate tool, what exactly is? Well, to certainly precisely tell what percent of your weight is fat, you need to be weighed underwater. That's a hassle, not the sort of thing the average person will do, certainly not on a regular basis. But, with just a tape measure, you are able to calculate the waist-hip ratio. Next you can utilize http://www.healthcalculators.org/calculators/waist_hip.asp to find if it's in a healthy range. This really is a much more sensible measure.