IMPACTO EN LAS PEQUEÑAS Y MEDIANAS EMPRESAS

The DCT is a special case of a discrete Fourier transform in which the sine components of the coefficients have been eliminated leaving a single number. This is actually quite easy. Figure 2.42(a) shows a block of input samples to a transform process. By repeating the samples in a time-reversed order and performing a discrete Fourier transform on the double-length sample set a DCT is obtained. The effect of mirroring the input waveform is to turn it into an even function whose sine coefficients are all zero. The result can be understood by considering the effect of individually transforming the input block and the reversed block. Figure 2.42(b) shows that the phase of all the components of one block are in the opposite sense to those in the other. This means that when the components are added to give the transform of the double length block, all of the sine components cancel out, leaving only the cosine coefficients, hence the name of the transform.3_{In practice the sine component} calculation is eliminated. Another advantage is that doubling the block length by mirroring doubles the frequency resolution, so that twice as many useful coefficients are produced. In fact a DCT produces as many useful coefficients as input samples. Clearly when the inverse transform is performed the reversed part of the waveform is discarded.

Figure 2.43 shows how a DCT is calculated by multiplying each sample in the input block by terms which represent sampled cosine waves of various frequencies. A given DCT coefficient is obtained when the result of multiplying every input sample in the block is summed. The DCT is primarily used in compression processing because it converts the input waveform into a form where redundancy can be easily detected and removed.

Figure 2.42 The DCT is obtained by mirroring the input block as shown in (a) prior to an FFT.

The mirroring cancels out the sine components as in (b), leaving only cosine coefficients.

The wavelet transform was not discovered by any one individual, but has evolved via a number of similar ideas and was only given a strong mathematical foundation relatively recently.2,4,5 _{The wavelet transform is} similar to the Fourier transform in that it has basis functions of various frequencies which are multiplied by the input waveform to identify the frequencies it contains. However, the Fourier transform is based on periodic signals and endless basis functions and requires windowing. The wavelet transform is fundamentally windowed, as the basis functions employed are not endless sine waves, but are finite on the time axis; hence the name. Wavelet transforms do not use a fixed window, but instead the window period is inversely proportional to the frequency being analysed. As a result a useful combination of time and frequency resolutions is obtained. High frequencies corresponding to transients in audio or edges in video are transformed with short basis functions and therefore are accurately located. Low frequencies are transformed with long basis functions which have good frequency resolution.

Figure 2.44 shows that a set of wavelets or basis functions can be obtained simply by scaling (stretching or shrinking) a single wavelet on the time axis. Each wavelet contains the same number of cycles such that as the frequency reduces the wavelet gets longer. Thus the frequency discrimination of the wavelet transform is a constant fraction of the signal frequency. In a filter bank such a characteristic would be described as ‘constant Q’. Figure 2.45 shows the division of the frequency domain by a wavelet transform is logarithmic whereas with the Fourier transform the division is uniform. The logarithmic coverage is effectively dividing the frequency domain into octaves and as such parallels the frequency discrimination of human hearing. For a comprehensive treatment of wavelets the reader is referred to Strang and Nguyen.6

As it is relatively recent, the wavelet transform has yet to be widely used although it shows great promise as it is naturally a multi-resolution transform allowing scalable decoding. It has been successfully used in audio and in commercially available non-linear video editors and in other fields such as radiology and geology.

2.20 Magnetism

Magnetism is vital to sound reproduction as it used in so many different places. Microphones and loudspeakers rely on permanent magnets, recording

Figure 2.45 Wavelet transforms divide the frequency domain into octaves instead of the equal

bands of the Fourier transform.

tape stores magnetic patterns and the tape is driven by motors which are driven by magnetism.

A magnetic field can be created by passing a current through a solenoid, which is no more than a coil of wire. When the current ceases, the magnetism disappears. However, many materials, some quite common, display a perma- nent magnetic field with no apparent power source. Magnetism of this kind results from the spin of electrons within atoms. Atomic theory describes atoms as having nuclei around which electrons orbit, spinning as they go. Different orbits can hold a different number of electrons. The distribution of electrons determines whether the element is diamagnetic (non-magnetic) or paramag- netic (magnetic characteristics are possible). Diamagnetic materials have an even number of electrons in each orbit, and according to the Pauli exclu- sion principle half of them spin in each direction. The opposed spins cancel any resultant magnetic moment. Fortunately there are certain elements, the transition elements, which have an odd number of electrons in certain orbits. The magnetic moment due to electronic spin is not cancelled out in these paramagnetic materials.

Figure 2.46 shows that paramagnetism materials can be classified as anti- ferromagnetic, ferrimagnetic and ferromagnetic. In some materials alternate atoms are antiparallel and so the magnetic moments are cancelled. In ferri- magnetic materials there is a certain amount of antiparallel cancellation, but a net magnetic moment remains. In ferromagnetic materials such as iron, cobalt or nickel, all of the electron spins can be aligned and as a result the most powerful magnetic behaviour is obtained.

Figure 2.46 The classification of paramagnetic materials. The ferromagnetic materials exhibit

the strongest magnetic behaviour.

It is not immediately clear how a material in which electron spins are parallel could ever exist in an unmagnetized state or how it could be partially magne- tized by a relatively small external field. The theory of magnetic domains has been developed to explain what is observed in practice. Figure 2.47(a) shows a ferromagnetic bar which is demagnetized. It has no net magnetic moment because it is divided into domains or volumes which have equal and opposite moments. Ferromagnetic material divides into domains in order to reduce its magnetostatic energy. Figure 2.47(b) shows a domain wall which is around 0.1µm thick. Within the wall the axis of spin gradually rotates from one state to another. An external field of quite small value is capable of disturbing the equilibrium of the domain wall by favouring one axis of spin over the other. The result is that the domain wall moves and one domain becomes larger at the expense of another. In this way the net magnetic moment of the bar is no longer zero as shown in Figure 2.47(c).

For small distances, the domain wall motion is linear and reversible if the change in the applied field is reversed. However, larger movements are irreversible because heat is dissipated as the wall jumps to reduce its energy. Following such a domain wall jump, the material remains magnetized after the external field is removed and an opposing external field must be applied which must do further work to bring the domain wall back again. This is a process of hysteresis where work must be done to move each way. Were it not for this non-linear mechanism magnetic recording would be impossible. If magnetic materials were linear, tapes would return to the demagnetized state immediately after leaving the field of the head and this book would be a good deal thinner.

Figure 2.48 shows a hysteresis loop which is obtained by plotting the magnetization M when the external field H is swept to and fro. On the

Figure 2.47 (a) A magnetic material can have a zero net moment if it is divided into domains

as shown here. Domain walls (b) are areas in which the magnetic spin gradually changes from one domain to another. The stresses which result store energy. When some domains dominate, a net magnetic moment can exist as in (c).

Figure 2.48 A hysteresis loop which comes about because of the non-linear behaviour of magnetic

achieve this is called the intrinsic coercive forcemHc. A small increase in the reverse field reaches the point where, if the field where to be removed, the remanent magnetization would become zero. The field required to do this is the remanent coercive force, rHc.

As the external field H is swept to and fro, the magnetization describes a major hysteresis loop. Domain wall transit causes heat to be dissipated on every cycle around the loop and the dissipation is proportional to the loop area. For a recording medium, a large loop is beneficial because the replay signal is a function of the remanence and high coercivity resists erasure. The same is true for a permanent magnet. Heating is not an issue.

For a device such as a recording head, a small loop is beneficial. Figure 2.49(a) shows the large loop of a hard magnetic material used for recording media and for permanent magnets. Figure 2.49(b) shows the small loop of a soft magnetic material which is used for recording heads and transformers.

According to the Nyquist noise theorem, anything which dissipates energy when electrical power is supplied must generate a noise voltage when in

Figure 2.49 The recording medium requires a large loop area (a) whereas the head requires a

thermal equilibrium. Thus magnetic recording heads have a noise mechanism which is due to their hysteretic behaviour. The smaller the loop, the less the hysteretic noise. In conventional heads, there are a large number of domains and many small domain wall jumps. In thin film heads there are fewer domains and the jumps must be larger. The noise this causes is known as Barkhausen noise, but as the same mechanism is responsible it is not possible to say at what point hysteresis noise should be called Barkhausen noise.