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Movie Title Year Distributor Notes Rev Formats 18 and Nasty 20 2001 Devil's Film 1 DRO Alter Ego 2002 Scarlet Tower Facial DRO Barely Legal 10 2000 Hustler Video Pee 3 DRO Bend Over Butts 2 2002 Digital Dreams O Bring 'um Young 4 2001 Anabolic Video Facial 5 DRO Chica Boom 11 2001 Kick Ass Pictures Facial 4 DRO Chillin' Wit The Mack 7 2003 Sean Michaels International LezOnly DO Coed Cocksuckers 22 2000 Zane Entertainment Group BJOnly Facial 1 DO Heavy Metal Teens 2010 Loaded Digital DRO Hi-teen Club 1 2002 Wildlife Facial Bald 2 DRO Hot Showers 3 2002 Hustler Video LezOnly 2 DRO Latin Extreme 1 2002 Jill Kelly Productions 1 DO Latina Mommasitas 2008 Pandemonium Facial Bald DO Legal Latinas 2006 Pandemonium Facial Bald DRO Little Latinas 2004 Maximum Xposure LezOnly DO Mamacitas 1 2002 Video Team LezOnly 2 DRO Naughty College School Girls 22 2002 New Sensations Facial 1 DRO
Real Sex Magazine 41 2001 New Sensations 1 DRO Smokin' 5 2002 Kick Ass Pictures MastOnly 2 DRO Spanish Fly Pussy Search 3 2001 Jake Steed Productions DRO Sticky Side Up 4 2002 West Coast Productions Facial 1 DRO Sweatin' It 1 2002 Kick Ass Pictures Facial Bald 4 DRO Taylor Wane's Slumber Party 2000 Dreamland Video LezOnly Bald 1 DO Teen Spirit 2 2002 Metro Facial 2 DRO Try-a-teen 14 2002 Visual Images Facial 3 DO University Coeds 30 2000 Dane Productions LezOnly 1 DO University Coeds 31 2001 Dane Productions LezOnly 1 DRO Viva Latina Ass 2 2008 DNA Facial Bald DO White Panty Chronicles 19 2001 Rain Productions LezOnly 1 D X-Cheerleaders Gone Fuckin' Nuts 2 2006 Nasty Jack DO XXX Raiders 2001 Relative Young and Tasty 2003 Filmco Releasing O Young Nasty and All Natural 1 2009 Wildlife Facial Bald O Young Sluts, Inc. 3 2001 Hustler Video



The bilinear transform (also known as 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. The bilinear transform is a special case of a conformal mapping (namely, a Möbius transformation), often used to convert a transfer function {\displaystyle H_{a}(s)\ }H_{a}(s)\ of a linear, time-invariant (LTI) filter in the continuous-time domain (often called an analog filter) to a transfer function {\displaystyle H_{d}(z)\ }H_{d}(z)\ of a linear, shift-invariant filter in the discrete-time domain (often called a digital filter although there are analog filters constructed with switched capacitors that are discrete-time filters). It maps positions on the {\displaystyle j\omega \ }j\omega \ axis, {\displaystyle Re[s]=0\ }Re[s]=0\ , in the s-plane to the unit circle, {\displaystyle |z|=1\ }|z|=1\ , in the 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 {\displaystyle \left(z^{-1}\right)\ }\left(z^{{-1}}\right)\ with first order all-pass filters. The transform preserves stability and maps every point of the frequency response of the continuous-time filter, {\displaystyle H_{a}(j\omega _{a})\ }H_{a}(j\omega _{a})\ to a corresponding point in the frequency response of the discrete-time filter, {\displaystyle H_{d}(e^{j\omega _{d}T})\ }H_{d}(e^{{j\omega _{d}T}})\ although to a somewhat different frequency, as shown in the 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. Contents 1 Discrete-time approximation 2 Stability and minimum-phase property preserved 3 General transformation of a continuous-time IIR filter 4 Example 5 Transformation of a first-order continuous-time filter 6 Transformation of a second-order biquad 7 Frequency warping 8 See also 9 References 10 External links Discrete-time approximation 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 unit impulse), the result is precisely the Z transform of the discrete-time sequence with the substitution of {\displaystyle {\begin{aligned}z&=e^{sT}\\&={\frac {e^{sT/2}}{e^{-sT/2}}}\\&\approx {\frac {1+sT/2}{1-sT/2}}\end{aligned}}}{\begin{aligned}z&=e^{{sT}}\\&={\frac {e^{{sT/2}}}{e^{{-sT/2}}}}\\&\approx {\frac {1+sT/2}{1-sT/2}}\end{aligned}} where {\displaystyle T\ }T\ is the numerical integration step size of the trapezoidal rule used in the bilinear transform derivation;[1] or, in other words, the sampling period. The above bilinear approximation can be solved for {\displaystyle s\ }s\ or a similar approximation for {\displaystyle s=(1/T)\ln(z)\ \ }s=(1/T)\ln(z)\ \ can be performed. The inverse of this mapping (and its first-order bilinear approximation) is {\displaystyle {\begin{aligned}s&={\frac {1}{T}}\ln(z)\\&={\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]\\&\approx {\frac {2}{T}}{\frac {z-1}{z+1}}\\&={\frac {2}{T}}{\frac {1-z^{-1}}{1+z^{-1}}}\end{aligned}}}{\begin{aligned}s&={\frac {1}{T}}\ln(z)\\&={\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]\\&\approx {\frac {2}{T}}{\frac {z-1}{z+1}}\\&={\frac {2}{T}}{\frac {1-z^{{-1}}}{1+z^{{-1}}}}\end{aligned}} The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function, {\displaystyle H_{a}(s)\ }H_{a}(s)\ {\displaystyle s\leftarrow {\frac {2}{T}}{\frac {z-1}{z+1}}.}s\leftarrow {\frac {2}{T}}{\frac {z-1}{z+1}}. That is {\displaystyle 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).\ }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).\ Stability and minimum-phase property preserved A continuous-time causal filter is stable if the poles of its transfer function fall in the left half of the 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 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. Likewise, a continuous-time filter is minimum-phase if the 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. General transformation of a continuous-time IIR filter Consider a continuous-time IIR filter of order {\displaystyle N}N {\displaystyle H_{a}(s)=k\prod _{i=1}^{N}{\frac {s-\xi _{i}}{s-p_{i}}},}{\displaystyle H_{a}(s)=k\prod _{i=1}^{N}{\frac {s-\xi _{i}}{s-p_{i}}},} where {\displaystyle p_{i}}p_{i} and {\displaystyle \xi _{i}}\xi _{i} are the transfer function poles and zeros in the s-plane. Let {\displaystyle K=2/T}{\displaystyle K=2/T} (or if using frequency warping as described below, let {\displaystyle K=\omega _{0}/\tan(\omega _{0}T/2)}{\displaystyle K=\omega _{0}/\tan(\omega _{0}T/2)}). The filter's bilinear transform is obtained by substituting {\displaystyle s=K(z-1)/(z+1)}{\displaystyle s=K(z-1)/(z+1)}: {\displaystyle {\begin{aligned}H_{d}(z)&=H_{a}{\bigl (}K{\tfrac {z-1}{z+1}}{\bigr )}\\&=k\prod _{i=1}^{N}{\frac {K{\frac {z-1}{z+1}}-\xi _{i}}{K{\frac {z-1}{z+1}}-p_{i}}}\\&=k\prod _{i=1}^{N}{\frac {K-\xi _{i}}{K-p_{i}}}\cdot {\frac {z-{\frac {K+\xi _{i}}{K-\xi _{i}}}}{z-{\frac {K+p_{i}}{K-p_{i}}}}}\\&=H_{a}(K)\prod _{i=1}^{N}{\frac {z-\xi _{i}^{d}}{z-p_{i}^{d}}},\end{aligned}}}{\displaystyle {\begin{aligned}H_{d}(z)&=H_{a}{\bigl (}K{\tfrac {z-1}{z+1}}{\bigr )}\\&=k\prod _{i=1}^{N}{\frac {K{\frac {z-1}{z+1}}-\xi _{i}}{K{\frac {z-1}{z+1}}-p_{i}}}\\&=k\prod _{i=1}^{N}{\frac {K-\xi _{i}}{K-p_{i}}}\cdot {\frac {z-{\frac {K+\xi _{i}}{K-\xi _{i}}}}{z-{\frac {K+p_{i}}{K-p_{i}}}}}\\&=H_{a}(K)\prod _{i=1}^{N}{\frac {z-\xi _{i}^{d}}{z-p_{i}^{d}}},\end{aligned}}} where {\displaystyle p_{i}^{d}}{\displaystyle p_{i}^{d}}, {\displaystyle \xi _{i}^{d}}{\displaystyle \xi _{i}^{d}} are the z-plane pole and zero locations of the discretized filter, {\displaystyle p_{i}^{d}={\frac {K+p_{i}}{K-p_{i}}},\quad \xi _{i}^{d}={\frac {K+\xi _{i}}{K-\xi _{i}}}.}{\displaystyle p_{i}^{d}={\frac {K+p_{i}}{K-p_{i}}},\quad \xi _{i}^{d}={\frac {K+\xi _{i}}{K-\xi _{i}}}.} Example As an example take a simple low-pass RC filter. This continuous-time filter has a transfer function {\displaystyle {\begin{aligned}H_{a}(s)&={\frac {1/sC}{R+1/sC}}\\&={\frac {1}{1+RCs}}.\end{aligned}}}{\begin{aligned}H_{a}(s)&={\frac {1/sC}{R+1/sC}}\\&={\frac {1}{1+RCs}}.\end{aligned}} If we wish to implement this filter as a digital filter, we can apply the bilinear transform by substituting for {\displaystyle s}s the formula above; after some reworking, we get the following filter representation: {\displaystyle H_{d}(z)\ }H_{d}(z)\ {\displaystyle =H_{a}\left({\frac {2}{T}}{\frac {z-1}{z+1}}\right)\ }=H_{a}\left({\frac {2}{T}}{\frac {z-1}{z+1}}\right)\ {\displaystyle ={\frac {1}{1+RC\left({\frac {2}{T}}{\frac {z-1}{z+1}}\right)}}\ }={\frac {1}{1+RC\left({\frac {2}{T}}{\frac {z-1}{z+1}}\right)}}\ {\displaystyle ={\frac {1+z}{(1-2RC/T)+(1+2RC/T)z}}\ }={\frac {1+z}{(1-2RC/T)+(1+2RC/T)z}}\ {\displaystyle ={\frac {1+z^{-1}}{(1+2RC/T)+(1-2RC/T)z^{-1}}}.\ }={\frac {1+z^{{-1}}}{(1+2RC/T)+(1-2RC/T)z^{{-1}}}}.\ 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. Transformation of a first-order continuous-time filter It is possible to relate the coefficients of a continuous-time, analog filter with those of a similar discrete-time digital filter created through the bilinear transform process. Transforming a general, first-order continuous-time filter with the given transfer function {\displaystyle H_{a}(s)={\frac {b_{0}s+b_{1}}{a_{0}s+a_{1}}}={\frac {b_{0}+b_{1}s^{-1}}{a_{0}+a_{1}s^{-1}}}}{\displaystyle H_{a}(s)={\frac {b_{0}s+b_{1}}{a_{0}s+a_{1}}}={\frac {b_{0}+b_{1}s^{-1}}{a_{0}+a_{1}s^{-1}}}} using the bilinear transform (without prewarping any frequency specification) requires the substitution of {\displaystyle s\leftarrow K{\frac {1-z^{-1}}{1+z^{-1}}}}s\leftarrow K{\frac {1-z^{{-1}}}{1+z^{{-1}}}} where {\displaystyle K\triangleq {\frac {2}{T}}}K\triangleq {\frac {2}{T}}. However, if the frequency warping compensation as described below is used in the bilinear transform, so that both analog and digital filter gain and phase agree at frequency {\displaystyle \omega _{0}}\omega _{0}, then {\displaystyle K\triangleq {\frac {\omega _{0}}{\tan \left({\frac {\omega _{0}T}{2}}\right)}}}{\displaystyle K\triangleq {\frac {\omega _{0}}{\tan \left({\frac {\omega _{0}T}{2}}\right)}}}. This results in a discrete-time digital filter with coefficients expressed in terms of the coefficients of the original continuous time filter: {\displaystyle H_{d}(z)={\frac {(b_{0}K+b_{1})+(-b_{0}K+b_{1})z^{-1}}{(a_{0}K+a_{1})+(-a_{0}K+a_{1})z^{-1}}}}{\displaystyle H_{d}(z)={\frac {(b_{0}K+b_{1})+(-b_{0}K+b_{1})z^{-1}}{(a_{0}K+a_{1})+(-a_{0}K+a_{1})z^{-1}}}} Normally the constant term in the denominator must be normalized to 1 before deriving the corresponding difference equation. This results in {\displaystyle H_{d}(z)={\frac {{\frac {b_{0}K+b_{1}}{a_{0}K+a_{1}}}+{\frac {-b_{0}K+b_{1}}{a_{0}K+a_{1}}}z^{-1}}{1+{\frac {-a_{0}K+a_{1}}{a_{0}K+a_{1}}}z^{-1}}}.}{\displaystyle H_{d}(z)={\frac {{\frac {b_{0}K+b_{1}}{a_{0}K+a_{1}}}+{\frac {-b_{0}K+b_{1}}{a_{0}K+a_{1}}}z^{-1}}{1+{\frac {-a_{0}K+a_{1}}{a_{0}K+a_{1}}}z^{-1}}}.} The difference equation (using the Direct Form I) is {\displaystyle y[n]={\frac {b_{0}K+b_{1}}{a_{0}K+a_{1}}}\cdot x[n]+{\frac {-b_{0}K+b_{1}}{a_{0}K+a_{1}}}\cdot x[n-1]-{\frac {-a_{0}K+a_{1}}{a_{0}K+a_{1}}}\cdot y[n-1]\ .}{\displaystyle y[n]={\frac {b_{0}K+b_{1}}{a_{0}K+a_{1}}}\cdot x[n]+{\frac {-b_{0}K+b_{1}}{a_{0}K+a_{1}}}\cdot x[n-1]-{\frac {-a_{0}K+a_{1}}{a_{0}K+a_{1}}}\cdot y[n-1]\ .} Transformation of a second-order biquad A similar process can be used for a general second-order filter with the given transfer function {\displaystyle H_{a}(s)={\frac {b_{0}s^{2}+b_{1}s+b_{2}}{a_{0}s^{2}+a_{1}s+a_{2}}}={\frac {b_{0}+b_{1}s^{-1}+b_{2}s^{-2}}{a_{0}+a_{1}s^{-1}+a_{2}s^{-2}}}\ .}{\displaystyle H_{a}(s)={\frac {b_{0}s^{2}+b_{1}s+b_{2}}{a_{0}s^{2}+a_{1}s+a_{2}}}={\frac {b_{0}+b_{1}s^{-1}+b_{2}s^{-2}}{a_{0}+a_{1}s^{-1}+a_{2}s^{-2}}}\ .} This results in a discrete-time digital biquad filter with coefficients expressed in terms of the coefficients of the original continuous time filter: {\displaystyle H_{d}(z)={\frac {(b_{0}K^{2}+b_{1}K+b_{2})+(2b_{2}-2b_{0}K^{2})z^{-1}+(b_{0}K^{2}-b_{1}K+b_{2})z^{-2}}{(a_{0}K^{2}+a_{1}K+a_{2})+(2a_{2}-2a_{0}K^{2})z^{-1}+(a_{0}K^{2}-a_{1}K+a_{2})z^{-2}}}}H_{d}(z)={\frac {(b_{0}K^{2}+b_{1}K+b_{2})+(2b_{2}-2b_{0}K^{2})z^{{-1}}+(b_{0}K^{2}-b_{1}K+b_{2})z^{{-2}}}{(a_{0}K^{2}+a_{1}K+a_{2})+(2a_{2}-2a_{0}K^{2})z^{{-1}}+(a_{0}K^{2}-a_{1}K+a_{2})z^{{-2}}}} Again, the constant term in the denominator is generally normalized to 1 before deriving the corresponding difference equation. This results in {\displaystyle H_{d}(z)={\frac {{\frac {b_{0}K^{2}+b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}+{\frac {2b_{2}-2b_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{-1}+{\frac {b_{0}K^{2}-b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{-2}}{1+{\frac {2a_{2}-2a_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{-1}+{\frac {a_{0}K^{2}-a_{1}K+a_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{-2}}}.}H_{d}(z)={\frac {{\frac {b_{0}K^{2}+b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}+{\frac {2b_{2}-2b_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{{-1}}+{\frac {b_{0}K^{2}-b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{{-2}}}{1+{\frac {2a_{2}-2a_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{{-1}}+{\frac {a_{0}K^{2}-a_{1}K+a_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}z^{{-2}}}}. The difference equation (using the Direct form I) is {\displaystyle y[n]={\frac {b_{0}K^{2}+b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot x[n]+{\frac {2b_{2}-2b_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot x[n-1]+{\frac {b_{0}K^{2}-b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot x[n-2]-{\frac {2a_{2}-2a_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot y[n-1]-{\frac {a_{0}K^{2}-a_{1}K+a_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot y[n-2]\ .}{\displaystyle y[n]={\frac {b_{0}K^{2}+b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot x[n]+{\frac {2b_{2}-2b_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot x[n-1]+{\frac {b_{0}K^{2}-b_{1}K+b_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot x[n-2]-{\frac {2a_{2}-2a_{0}K^{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot y[n-1]-{\frac {a_{0}K^{2}-a_{1}K+a_{2}}{a_{0}K^{2}+a_{1}K+a_{2}}}\cdot y[n-2]\ .} Frequency warping To determine the frequency response of a continuous-time filter, the transfer function {\displaystyle H_{a}(s)}{\displaystyle H_{a}(s)} is evaluated at {\displaystyle s=j\omega _{a}}{\displaystyle s=j\omega _{a}} which is on the {\displaystyle j\omega }{\displaystyle j\omega } axis. Likewise, to determine the frequency response of a discrete-time filter, the transfer function {\displaystyle H_{d}(z)}{\displaystyle H_{d}(z)} is evaluated at {\displaystyle z=e^{j\omega _{d}T}}{\displaystyle z=e^{j\omega _{d}T}} which is on the unit circle, {\displaystyle |z|=1}{\displaystyle |z|=1}. The bilinear transform maps the {\displaystyle j\omega }{\displaystyle j\omega } axis of the s-plane (of which is the domain of {\displaystyle H_{a}(s)}{\displaystyle H_{a}(s)}) to the unit circle of the z-plane, {\displaystyle |z|=1}{\displaystyle |z|=1} (which is the domain of {\displaystyle H_{d}(z)}{\displaystyle H_{d}(z)}), but it is not the same mapping {\displaystyle z=e^{sT}}{\displaystyle z=e^{sT}} which also maps the {\displaystyle j\omega }{\displaystyle j\omega } axis to the unit circle. When the actual frequency of {\displaystyle \omega _{d}}{\displaystyle \omega _{d}} is input to the discrete-time filter designed by use of the bilinear transform, then it is desired to know at what frequency, {\displaystyle \omega _{a}}{\displaystyle \omega _{a}}, for the continuous-time filter that this {\displaystyle \omega _{d}}{\displaystyle \omega _{d}} is mapped to. {\displaystyle H_{d}(z)=H_{a}\left({\frac {2}{T}}{\frac {z-1}{z+1}}\right)}{\displaystyle H_{d}(z)=H_{a}\left({\frac {2}{T}}{\frac {z-1}{z+1}}\right)} {\displaystyle H_{d}(e^{j\omega _{d}T})}{\displaystyle H_{d}(e^{j\omega _{d}T})} {\displaystyle =H_{a}\left({\frac {2}{T}}{\frac {e^{j\omega _{d}T}-1}{e^{j\omega _{d}T}+1}}\right)}{\displaystyle =H_{a}\left({\frac {2}{T}}{\frac {e^{j\omega _{d}T}-1}{e^{j\omega _{d}T}+1}}\right)} {\displaystyle =H_{a}\left({\frac {2}{T}}\cdot {\frac {e^{j\omega _{d}T/2}\left(e^{j\omega _{d}T/2}-e^{-j\omega _{d}T/2}\right)}{e^{j\omega _{d}T/2}\left(e^{j\omega _{d}T/2}+e^{-j\omega _{d}T/2}\right)}}\right)}{\displaystyle =H_{a}\left({\frac {2}{T}}\cdot {\frac {e^{j\omega _{d}T/2}\left(e^{j\omega _{d}T/2}-e^{-j\omega _{d}T/2}\right)}{e^{j\omega _{d}T/2}\left(e^{j\omega _{d}T/2}+e^{-j\omega _{d}T/2}\right)}}\right)} {\displaystyle =H_{a}\left({\frac {2}{T}}\cdot {\frac {\left(e^{j\omega _{d}T/2}-e^{-j\omega _{d}T/2}\right)}{\left(e^{j\omega _{d}T/2}+e^{-j\omega _{d}T/2}\right)}}\right)}{\displaystyle =H_{a}\left({\frac {2}{T}}\cdot {\frac {\left(e^{j\omega _{d}T/2}-e^{-j\omega _{d}T/2}\right)}{\left(e^{j\omega _{d}T/2}+e^{-j\omega _{d}T/2}\right)}}\right)}


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