Linear Distortions

Phase- and Amplitude errors cause linear distortions

inear distortions do not produce any new frequencies contrary to nonlinear distortions. Let it me descriptive in pictures and simple words explain, without the assistance of the mathematically oriented filter theory, of it gives already it to enough literature, which serves however often only for the self-manifestation of the mathematical garbness of the author and which practical meaning often neglects. This literature is often meaningful for the beginner only if it understood used mathematics already before.

Reasons

Phase Failures - each amplifier necessarily for a certain frequency a signal running time, i.e., it passes a finite time, until at the output of the amplifier the signal is available again. This turn-around time is finally, shorter with a fast amplifier naturally than with slower ones. The signal propagation delay time will with sinusoidal signals indirectly measurably as phase shift, it becomes visibly by the representation of input and output signal common on an oscilloscope. With increasing signal frequency the phase shift continues to rise between input and output signal. Now e.g. if a music signal is set on an amplifier, then these different frequencies amplifies with different amplitude errors and phase shifts. The result is a signal from the amplifier, which looks no longer in such a way like the original signal, it is not only amplified (which it should also be), but it is also additionally linear distorted.

Amplitude error - due to the finite bandwidth of an amplifier, cannot transfer the amplifier in particular higher-frequency signals any more 1 to 1, i.e. an amplitude from e.g.. 1 V is transferred only with 0.98 V. If the signal frequency reached the bandwidth, then only 0.707 V become from 1 V. Now even if a music signal is set here on the amplifier, then these are amplified differently frequencies with different amplification factors. The signal from the amplifier is not comparable any longer with the original signal. Linear distortions developed in such a way.

Combination from both sources of error - both sources of error come combined forwards. A too low bandwidth of the amplifier related to the used signal frequencies, causes clear amplitude and phase errors.

For an Audio Amplifier - a sufficiently high bandwidth  is to be obtained for a hifi amplifier of elementary importance. The linear distortions remain so small. Also each sound control (bass, treble) produces linear distortions. For example an treble control raises high frequencies and causes thereby a desired amplitude error. Normally a sound controll should be only used, in order to correct small amplitudes again. In this case by inserting linear distortions the entire amplitude response is again linearized. Note: e.g. a treble control causes the desired amplitude error, in addition, automatically an inadvertent phase error, which falsify again the original signal. Those are fundamental filter characteristics. Sound controllers are from electrotechnical view only sources of error, which cause that the way to ideal amplifier will leave.

Naturally there are voices - those maintains the human ear is substantially more sensitive for amplitude errors than for phase errors. My experience confirms this clearly also. However to state phase errors are uninteresting, that are also not o.k. Is often tried to define a low bandwidth of a Hifi amplifier as less important and to play the importance down of the amplifiers bandwidth. When phase errors become audible, I cannot discuss, since I did not undertake detailed attempts themselves ever and also do not need. A loudspeaker manufacturer has herein very much more experience with "whether" "when" and "how" linear distortions become audible. With its crossover networks, housings, rooms and loudspeakers he must concern himself very much with phase relationships and their effects.

Honestly - "whether" "when" and "how" it is for me less of importance, I tries these linear distortions by good circuits to avoid from the beginning. Distortions behind a rhetorical curtain to hide is not the fine kind. It is an incontestable fact, linear distortions is from electrotechnical view a clear error, which it applies to avoid.

Make Linear Distortions visible

On the basis a descriptive simple mathematical simulation the occurrence of linear distortions is to be demonstrated. In addition signal frequencies, amplitudes and phase errors were accepted. It proceeded from an amplifier, which has a bandwidth of approximately only 15 kHz. An amplification factor of "unity" simplifies the simulation.

Definition

Figure 1 (equations left side)

Simulation from time zero to  2 milliseconds

Signalfrequencies f1 to f4. 

Original signal amplitudes A to D. 

Real amplitudes with amplitude errors AA to DD.

The original signal consists of four sine waves voltages a(t) to d(t).

The time domain of the original signal sum(t)

The original is applied to the input of an simulated bad slow amplifier. Arbitrary assumption:

at 1 kHz only 0.99 volts instead of 1 volt (-1%)

at 2.53 kHz 0.71 voltv instead of 0.7 volt (+1,4%) less overshooting

at 9.71 kHz 0.47 volts instead of 0.5 volt (-6%)

at 14.3 kHz 0.25 volts instead of 0.3 volt (-17%)

The adjusted phase errors in the argument of the sinefunction, 5, 15, 25, and 45 degrees.

The signals aa(t) to dd(t) showing the changed single signals.

The signal sum (t) is the  originalsignal of an ideal audio amplifier

The singal sumsum(t) is the relistic signal of the simulated bad audio amplifier.

The model is easily repeatable for everyone with a mathematics program.

Input signals applied on amplifier

Figure 2 shows the orginal signals, which are set in each case on the simulated amplifier.

Output signals of amplifier

Figure 3 shows the output signals of the simulated amplifier.

Comparison of original signal and simulated signal

Figure 4 shows the additive compound signal sum(t) in red, as well as the simulated, the amplifier passed output signal sumsum(t) in blue. To see clear, the output signal limps to the original signal, due temporally afterwards to the phase errors. Also recognizably the output signal is with the highest frequencies a little sanded, due to the amplitude errors.

Here consciously the frequency response of a very bad hifi amplifier was taken, in order to show also in the diagram the errors clearly. With an ideal amplifier the blue curve should be directly congruently over the red curve. Whether would be now actually audible, is another history, which is to judge everyone. What says your feeling to you? Electrotechnically it is already clearly no more the ideal case accordingly.

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