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Positive and Negative Logic Usually, the binary digit 1 stands for true and is represented by a voltage of about +5 V The binary digit 0 stands for false and is represented by about 0 V This is positive logic There are

Completing the job, we check (3, 31) in that equation: 3x 2 + y + 5x 11 = 0 3 32 + ( 31) + 5 3 11 = 0 3 9 + ( 31) + 15 11 = 0 27 + ( 31) + 15 11 = 0 4 + 15 11 = 0 11 11 = 0 0=0

The graph of Fig 26-12D illustrates a method of analog-to-digital (A/D) conversion A voice signal, or any continuously variable signal, can be digitized, or converted into a string of pulses, whose amplitudes can achieve only a finite number of states

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You might ask, Is it possible for a two-by-two system, in which one or both of the equations is quadratic, to have imaginary or complex solutions The answer is Yes In that case, when we use the quadratic formula, those solutions will appear You ll see this happen in later in this chapter

other logic forms, the most common of which is negative logic In a common form of negative logic, the digit 1 (the logic high state) is represented by about 0 V, and the digit 0 (the logic low state) is represented by about +5 V The remainder of this chapter deals with positive logic

The number of states is always a power of 2, so that it can be represented as a binary-number code Fidelity gets better as the exponent increases The number of states is called the sampling resolution, or simply the resolution You might think that the resolution would have to be very large for good reproduction to be possible But in fact, a resolution of 23 = 8 (as shown in Fig 26-12D) is good enough for voice transmission, and is the standard resolution for commercial digital voice circuits A resolution of 24 = 16 is adequate for compact disks used in advanced hi-fi systems!

Here s a challenge!

Suppose that a1, a2, b1, b2, c1, and c2 are real numbers, and neither a1 nor a2 is equal to 0 Consider these two functions of x : y = a1x 2 + b1x + c1 and y = a2x 2 + b2x + c2 Derive a general formula for solving this system

The efficiency with which a signal can be digitized depends on the frequency at which sampling is done In general, the sampling rate must have a frequency that is at least twice the highest data frequency For an audio signal with components as high as 3 kHz, the minimum sampling rate for effective digitization is 6 kHz, or one sample every 167 microseconds ( s) Ideally, the sampling rate should be somewhat higher; the commercial voice standard is 8 kHz, or one sample every 125 s For music and hi-fi digital transmission, the standard sampling rate is 441 kHz, or one sample every 227 s This is based on a maximum audio frequency of 20 kHz, the approximate upper limit of the human hearing range

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