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The truth of eq. 9-A can be established as follows. let meaning that raise both sides to power n by definition hence; since y log x log x y x 10y xn 10yn log xn yn log xn n log x
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thus proving eq. (9-A). Let us now make use of eq. (9-A) as follows. Recall that, in a purely resistive circuit, POWER can be calculated by the equation V 2 =R. Thus, for two voltages, V1 and V2 , applied to a resistance of R ohms, the POWER RATIO is P
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2 V1 =R V1 =V2 2 2 V2 =R
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and thus, substituting this value of P in eq. (315) then making use of eq. (9-A), eq. (315) becomes dB 20 log V1 =V2 10-A
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which, it should be noted, is true only if V1 and V2 are both applied to the same value of resistance of R ohms. Actually, however, in practice eq. (10-A) is often applied in cases where the two resistances are not equal; in such a case the results are not really in decibels but in what we could call logarithmic units.
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Phase (Time-Delay) Distortion
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It s been pointed out that frequency discrimination (frequency distortion) is produced by the presence of AMPLITUDE DISTORTION and TIME-DELAY (PHASE) DISTORTION. First, in regard to amplitude distortion, it s clear that no amplitude distortion can occur if, in passing through a network, the amplitudes of all frequency components are multiplied by a xed constant value k; that is, if all frequency components are treated the same, as far as amplitudes are concerned. Let us, therefore, turn our attention to time-delay distortion, as follows. Since there is energy storage associated with inductance and capacitance, it s understandable that time is required to change the state of energy level in these parameters. Because of this, a time delay exists between the input and output waves of voltage and current in a network. If such time delay is the same for all frequency components, then there is no distortion due to time delay. This is illustrated in Figs. 27-A and 28-A, in which the INPUT signal, Fig. 27-A, consists of fundamental, second-harmonic, and fourth-harmonic waves, having amplitudes and positions as shown, the independent variable being time, t. Now let T denote the time delay between input and output waves, and suppose we have the desired condition in which T is the same for all frequency components, which is the condition illustrated in Fig. 28-A. Comparison of the two gures makes clear that if there is no amplitude distortion, and if the TIME DELAY is the SAME FOR ALL FREQUENCIES, then the output wave will be delayed, relative to the input wave, by T seconds, but the basic WAVESHAPE of the output wave will be the SAME as that of the input wave.
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Fig. 27-A. INPUT wave.
Fig. 28-A. OUTPUT wave.
As already noted, time-delay distortion is also called phase distortion. If we choose to talk in terms of phase distortion, the two conditions for distortionless transmission through a network are that (1) there be no amplitude distortion, and (2) a linear ( rst degree) relationship exist between phase shift and order of harmonic in the output wave. The meaning of this statement can be explained as follows. We recall, from note 18, that any non-sinusoidal repetitive waveform can be expressed as the sum of a fundamental sinusoidal wave and its harmonics, in which the amplitudes of the harmonics decrease in a general way as the order of the harmonic increases. With this in mind, let vi denote the instantaneous value of such a voltage waveform applied to the INPUT of a network, and let the Fourier series for vi be of the form vi a1 sin !t a2 sin 2!t an sin n!t 11-A where a1 sin !t is the fundamental (lowest frequency) component, a2 sin 2!t is the secondharmonic component, and so on, to any nth harmonic component of amplitude an and frequency n!. Now suppose, in passing through the network, that all the amplitudes are multiplied by the same constant value k (thus no amplitude distortion), and that all the components are delayed by the same amount of T seconds. In such a case, upon setting ka1 b1 ; ka2 b2 , and so on, and upon replacing t with t T),* eq. (11-A) becomes the OUTPUT VOLTAGE of the network vo b1 sin !t !T b2 sin 2!t 2!T bn sin n!t n!T 12-A
Equation (12-A) is true for the ideal condition in which all frequency components are delayed the same amount of time, T seconds, in passing through a network. Thus an sin n!t, applied at the INPUT, appears at the OUTPUT as bn sin n!t n bn sin n!t n!T
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