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Thanks to the work performed by Harry Nyquist at Bell Laboratories in the 1920s, we know that to optimally represent an analog signal as a digitally encoded bitstream, the analog signal must be sampled at a rate that is equal to twice the bandwidth of the channel over which the signal is to be transmitted. Since each analog voice channel is allocated 4 KHz of bandwidth, it follows that each voice signal must be sampled at twice that rate, or 8,000 samples per second. In fact, that is precisely what happens in T-Carrier systems, which we now use to illustrate our example. The standard T-Carrier multiplexer accepts inputs from 24 analog channels, as shown in Figure 3-66. Each channel is sampled, in turn, every one eight-thousandth of a second in round-robin fashion, resulting in the generation of 8,000 pulse amplitude samples from each channel every second. The sampling rate is important. If the sampling rate is too high, too much information is transmitted and bandwidth is wasted; if the sampling rate is too low, then we run the risk of aliasing. Aliasing is the interpretation of the sample points as a false waveform, due to the paucity of samples. This PAM process represents the first stage of PCM, the process by which an analog baseband signal is converted to a digital signal for transmission across the T-Carrier network. This first step is shown in Figure 3-67. The second stage of PCM, shown in Figure 3-68, is called quantization. In quantization, we assign values to each sample within a constrained range. For illustration purposes, imagine what we now have before us. We have replaced the continuous analog waveform of the signal with a series of amplitude samples which are close enough together that we can

The free cash flow to the firm approach takes net operating profit, estimates an adjusted tax payment based on the corporation s income tax rate, and subtracts adjusted taxes to calculate NOPAT. We call them adjusted tax payments because the actual tax payments to the various governmental taxing entities are arrived at through complex calculations following arcane tax laws, with the associated $600 per hour New York tax lawyer s fee tacked on. These calculations include deferred components and complicated tax code treatments that a regular person can t even begin to understand. Sometimes, due to taking advantage of tax laws, a company s tax rate is significantly below the maximum marginal corporate tax rate which in the United States is 35 percent. At other times, due to previous deferrals, the company s tax rate may be significantly above the maximum marginal tax rate. If the company remains profitable, in the long run the tax rate should approach or average close to the maximum marginal tax rate. The tax deductions for interest expense associated with debt and the tax shield that debt provides are not included in the cash flow section of the FCFF approach. They are introduced into the FCFF valuation process by adjusting the weighted average cost of capital. These adjustments, which are described in 6, reduce the cost of the debt component of the WACC to convert it to an after-tax WACC. The FCFF approach to calculate NOPAT is to multiply NOP by (1 Tax Rate). NOPAT NOP TAXES NOP * (1 Tax Rate)

Figure 3-66 Time-division multiplexing (TDM)

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discern the shape of the original wave from their collective amplitudes. Imagine also that we have graphed these samples in such a way that the wave of sample points meanders above and below an established zero point on the x-axis, so that some of the samples have positive values and others are negative, as shown. The amplitude levels allow us to assign values to each of the PAM samples, although a glaring problem with this technique should be obvious to the careful reader. Very few of the samples actually line up

If you believe that the company will become or continue to be profitable, will deplete their accelerated deductions, and will eventually pay taxes at the maximum marginal corporate tax rate, you may want to look at a what-if scenario that uses the maximum corporate tax rate, currently 35 percent, in the valuation calculation. That s a simple exercise if you are using DCF software such as ValuePro 2002.

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V=V(14) > 0 10 : 0

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We find Cisco s tax rate from its income statement by dividing its income before provision for income taxes by its provision for income

exactly with the amplitudes delineated by the graphing process In fact, most of them fall between the values, as shown in the illustration It doesn t take much of an intuitive leap to see that several of the samples will be assigned the same digital value by the coder-decoder that performs this function, yet they are clearly not the same amplitude This inaccuracy in the measurement method results in a problem known as quantizing noise, and is inevitable when linear measurement systems, such as the one suggested by the drawing, are employed in coder-decoders (CODECs) Needless to say, design engineers recognized this problem rather quickly, and equally quickly came up with an adequate solution.

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