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L = α f s δ f

where L , f s , and δ f are as just defined, and α is given by

α = 0 . 22 + 0 . 0366 · S B R

The validity of these simplified formulas depends on a number of assumptions, detailed in [link] , but all of them are sufficiently satisfied in this case to permit accuracy in the estimation of L within 5% or so.

Examination of [link] shows that δ f , the filter transition band, can be no larger than Δ f - B , the difference between the channel spacing and the bandwidth of each channel. Recalling also that N · Δ f = f s , we find that

L = N α Δ f δ f = N α 1 1 - B Δ f .

Thus, to first order, the pulse response duration of the required filter is proportional to the number of channels N and is hyperbolic in the percentage bandwidth , the ratio of the channel bandwidth B to the channel spacing Δ f . The effect of the proportionality to α will be examined shortly.

This graph consist of a horizontal line labeled Frequency spanning the length of the image with its extremes labeled -f_s/2 on the left and  f_s/2 on the right. There are five right triangles spaced at equal distances from each other sitting on the line. On top of the line there is also a waveform running across the horizontal line. The waveform continues past the first two triangles and then the wave and then ends at a large right triangle next to the middle smaller right triangle. On the other side of this triangle is another large right triangle that is turned the opposite direction with its right angle on the bottom left. The expanse between the far right of the large triangle of the left and the far left of the right triangle is labeled below the horizontal line and is labeled B. In the middle of the small middle triangle is a line that extends vertically and then intersect a line that connects the top corners of the large triangles. The middle line extending from the center circle is also crosses two parallel horizontal lines. An arrow points to these horizontal lines and labels it Passband Ripple PBR. The expanse between the beginning and the end of the right large triangle is labeled below the horizontal line as sf. The expanse from the center of the line to the middle of the second little triangle from the right. it is labeled delta f. On the right side of the large right triangle the waveform begins again. Between the right two triangles is labeled Stopband Ripple SBR.
Overlay of the Required Tuner Filter with the Generalized Response of an Optimal Linear Phase Equal-Ripple FIR Filter

Relationship to the design parameter Q

The development presented in the section Derivation of the equations for a Basic FDM-TDM Transmux defined the integer variable Q as the ratio of L and N . It was pointed out there without proof that in fact Q was an important design parameter, not just the artifact of two others. This can now be seen by combining the relationship L Q N with [link] to produce an expression for Q :

Q = α Δ f δ f = α 1 1 - B Δ f

Since N depends strictly on the number of channels into which the input band is divided, Q contains all of the information about the impact of the desired filter characteristics.

Continuation of the telegraphy demodulation example

Consider again the example of demodulating R.35 FDM FSK VFT canals discussed in the section Example: Using an FDM-TDM Transmux to Demodulate R.35 Telegraphy Signals . In that section, we determined that the following parameters would be appropriate: f s = 3840 Hz, N = 64 , and Δ f = 60 Hz. To determine Q , and hence the rate of computation needed for the data weighting segment of the transmultiplexer, we need to specify B and S B R , the degree of stopband suppression required.

Generally speaking, the filters in an FSK demodulator need to have unity gain at the mark or space frequency and zero gain at the space or mark frequency, respectively. A computer simulation used to verify the design of the demodulator showed that suppression of 50 dB was more than enough to provide the needed performance. At first glance it might appear that the transition band δ f can be allowed to equal the tone spacing Δ f = 60 Hz, making the percentage bandwidth equal to zero. Actual FSK VFT systems, however, sometimes experience bulk frequency shifts of several Hertz. In order to maintain full performance in the presence of such frequency offsets, the tuner filters need to be designed with a passband bandwidth of 15 Hz or so. Using S B R = 50 dB in [link] , we find with [link] that the required value of Q for this application is about 2.71. The actual value chosen for this application was 3, producing a pulse response duration of L = Q N = 192 , with the remaining degrees of freedom in the filter design used to widen the filter still more, allowing for even more frequency offset.

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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, An introduction to the fdm-tdm digital transmultiplexer. OpenStax CNX. Nov 16, 2010 Download for free at http://cnx.org/content/col11165/1.2
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