Manual Mathematical Principles of Signal Processing: Fourier and Wavelet Analysis

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For fixed parameters the model is linear. It is interpreted as a linear approximation to the process at these fixed parameter values. Wavelets give rise to linear transformations. The description of signal processing in the cochlea by wavelet transformations, where the wavelets depend on parameters, is compatible with this approach. This is the transfer function that is defined from the response of the linear system to pure sounds.

To an input signal. By choosing an appropriate scale on the x -axis, the multiple can be taken to be 1. The scaling law is then expressed as. Intimately connected to scaling is the concept of a tonotopic order. It is a central feature in the structure of the auditory pathway.

Frequencies of the acoustic signal are associated to places, at first in the cochlea and in the following stages in the various neuronal nuclei. The assignment is monotone, it preserves the order of the frequencies. Its inverse is called the tonotopic axis. The characteristic frequency CF is then the low level limit of the best frequency. The constant K is determined by inserting a special value for x. This is not strictly true, but it simplifies the exposition.

In subsequent sections a general theory will be developed that incorporates quite general scaling behavior. With the availability of advanced experimental data Rhode [ 6 ], Kiang and Moxon [ 7 ], Liberman [ 8 ], [ 9 ], Eldredge et al. Shera [ 12 ] gives the formula. The scaling variable that goes with it is.

The constant S is referred to as the shift. In the present treatment the frequency localization of a function will be defined as an expectation value in the frequency domain. The response to a general signal f t with Fourier representation. This leads to the equivalent formulation. This is recognized as a wavelet transform. The fact, that the cochlea - in a first approximation - performs a wavelet transform appears in the literature in , both in [ 1 ] and in [ 2 ]. It derives from the scaling symmetry in combination with time-invariance.

In addition, there is the circle group S that is related to phase shifts. Its action commutes with the action of the affine group. For this group, the uncertainty principle can be formulated. The functions for which equality holds in the uncertainty inequalities are called the extremal functions. They play a special role, similar as in quantum physics the coherent states the extremals for the Heisenberg uncertainty principle. The starting point in the present work is the tenet that these functions provide an approximation for the cochlear transfer function.

That the extremal functions should play a special role is not a new idea. In a paper by Irino [ 15 ] the idea is taken up in connection with signal processing in the cochlea. It is further developed by Irino and Patterson [ 16 ] in The presentation in this paper is based on previous work Reimann, [ 17 ]. The concept pursued is to determine the extremals in the space of real valued signals and to use a setup in the frequency domain, not in the time domain.

Different representations of the affine group give different families E c of extremal functions. The parameter c is used to adjust to the sound level and hence to provide linear approximations at different levels to the non-linear behavior of cochlear signal processing. The basic uncertainty inequalities for this group are then explicitly derived. The analysis builds on previous results Reimann [ 17 ]. A modification is necessary because the treatment of the phase in [ 17 ] was not satisfactory. This term comes in naturally and it will influence the argument - but not the modulus - of the extremal functions associated to the uncertainty inequalities.

The extremal functions for the basic uncertainty principle are interpreted as the transfer function at high levels of sound. With increasing parameter values the extremal functions for the general uncertainty inequality are then taken as approximations to the cochlear response at decreasing levels of sound. This group also acts directly in frequency space:. The induced unitary action on L 2 R , C is.

With this convention, the group action and the induced action are denoted with the same symbol. Clearly, the invariance property of the basilar membrane transfer function directly reflects this group action. Of relevance to our considerations is the space L 2 R , R of real valued signals of finite energy. Under the Fourier transform it is mapped onto.

The only action that commutes with both of them is the action. The basic variables in cochlear signal processing are time t and position x along the cochlea. In our approach the tonotopic axis is given by the exponential law. The intertwining action. This leads to the new inequality. This inequality is of the same nature as the previous inequality. It has the interpretation of an expectation value for the frequency. Later it will be associated with the place along the cochlea.

This is treated in [ 17 ]. A function h is called extremal, if equality holds for it in the uncertainty relation. The extremal functions are expected to play a special role in the signal processing of the cochlea. At the outset of the present discussion is however the fact that the cochlea performs a wavelet transform - and not a Fourier transform. It should therefore be expected that the extremal functions as discussed below play the crucial role in the hearing process. The tenet is now:. Note further that. The question then arises whether the experiments confirm the tenet.

Understanding Wavelets, Part 1: What Are Wavelets

To arrive at a preliminary conclusion, graphs of the modulus and of the real part of the function h are displayed in Figure 1. The extremal function on a relative scale. The extremal function is shown for a fixed frequency f as a function of the distance d in mm to the stapes. However the situation is of course not so simple. The basic problem is the non-linearity of the process that associates the movement u x , t of the basilar membrane to the input signal f t. This process is highly compressive and therefore its description by a transfer function can at best be looked at as an approximation.

The outcome can be compared to the experimental results obtained with life animals, yet at high intensities of sound pressure. The above description of the basilar membrane transfer function is therefore taken to be a linear approximation at high levels of sound pressure. In the following section the approach will be modified with the aim of obtaining linear approximations at all levels of sound pressure. There is the single non trivial commutator relation:. In fact the inequality is true for both variants and in both cases, families of inequalities depending on the parameters are obtained.

The question is how the extremal functions that are associated to these inequalities vary with the parameters. The extremal functions are obtained from the relation. Its choice is arbitrary. This gives a possibility for fine adjustment of the extremal function h c that describes the linear approximation at level c of the basilar membrane filter. The parameter c allows to express at which level of sound intensity the linearization is specified. Experimental results on the basilar membrane transfer function are reviewed in Robles and Rugggero, [ 18 ].

As already pointed out in the previous paper [ 17 ], the shape of the modulus of the transfer function determined at various intensities as given by Rhode and Recio [ 19 ], Figure 1C is approximately described by the modulus of the extremal functions at the corresponding parameter values. In particular, at a fixed position, the modulus of the transfer function has its peak below the position of the frequency localization and with decreasing intensity of sound level it approaches this position.

With the present setup the argument of the basilar membrane filter is independent of c. The experimental results by Rhode and Recio show minor changes of phase in dependence of the intensity level. With increasing intensity there is a small phase lag below the characteristic frequency and an equally small phase lead for frequencies above the characteristic frequency. Studies of the impulse response also confirm that the phase is almost invariant under changes in sound level Recio and Rhode [ 20 ], Shera [ 21 ].

The phase of the extremal function does not satisfy this requirement because of the logarithmic term. Yet still, the phase of the extremal function serves as an approximation of the physiological phase function on the interval in which the the absolute value is relevant. At places at which the absolute value is close to zero, the argument is of no significance. In Figure 3 the phase is pictured as a function of frequency in Hz. In this figure, the characteristic frequency is 7, Hz.

The part above about 3, Hz is of physiological relevance. The approximation holds in this range. It should be compared with the experimental results by Rhode and Recio [ 19 ], Figure 2E. The part below 3, Hz is the mathematical expression for the phase function. It is physiologically not correct, but this is of no significance. Phase as a function of d.

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The phase in cycles of the extremal function h as a function of the distance d in mm to the stapes. The frequency is Hz. Phase as a function of f. The solid line is the phase as determined by the extremal function. The dashed line is the physiologically correct substitute at low frequencies. Note that the phase values in this region are practically irrelevant for signal processing, because the amplitude values in this region are negligible. The parameters are the same as in Figure 2. They are the extremal functions for the uncertainty inequality.

Extremals for the uncertainty principle satisfy differential equations. Since the membrane transfer function is described by an extremal function and its transforms under the symmetry group and since the extremal functions are preserved under this action, it is possible to derive differential equations for the output of the signal. The resulting equations are called the structure equations.

In this situation differentiation of the wavelet transform W f a , t with respect to the parameters of the symmetry group directly leads to a differential equation. A linearization process for the kernel brings it back to a differential equation that is then satisfied approximately. The quantities in the equation at first are derivatives of the output function W f a , t and its Hilbert transform. A further calculation then shows that the result can be formulated as an inhomogeneous system of linear partial differential equations for the phase and for the logarithm of the amplitude of the output signal.

This is particularly satisfying because these are exactly the physiologically relevant quantities. The wavelet transform is then. The normalized extremal function h satisfies the differential equation. This gives the basic equation. This transform is a unitary operator on L 2 R , C. It extends to a bigger class of functions to all temperate distributions. On the basic trigonometric functions it operates very simply:. It immediately follows that. The linearity assumption then implies that this holds for arbitrary input signals f.

Fourier analysis - Wikipedia

The Hilbert transform thus appears naturally in this setting. Taking the factor a into account this is. Notice the shift by 1 2 that has its origin in the factor 1 a. Only the constants are slightly different. The structure equations can be written in x , t -coordinates:. Signal processing in the cochlea is non-linear. The main - but certainly not the only - source of non-linearity is the compressive nature inherent in the hearing process. In the abstract model pursued here this is taken care of with a single parameter that represents the level of sound intensity.

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  6. The model then describes the linear approximations at these levels. The structure equations are at the core of this abstract model, in fact they comprise all the essential features. First of all, they are linear as would be expected from a linear approximation. From a mathematical point of view, the equations therefore are very simple. On top, the system is quite special. Its solutions can be realized in complex form as products of two factors, the first of which is entirely determined by the system and the second is a holomorphic function that can be calculated from the signal.

    At every level c it is thus possible to associate to an input signal in a unique way a holomorphic function that describes the output signal in terms of the physiological parameters. The phase and the logarithm of the amplitude are used in the description of the experiments and they are omnipresent in all the representations of the auditory pathway. In themselves they are of limited significance, because they are not coded as such. What really is essential in any cochlear or in any neural model are the changes of these quantities, both with respect to time and with respect to the place.

    The structure equations precisely relate the local and temporal derivatives of phase and logarithm of amplitude. The geometry of the cochlea implicitly is inherent in the extremality property of the basilar membrane filter. But in the structure equations this only shows in terms of the constants. The implicit appearance of the tonotopic axis is an expression of the basic invariance principle that stands at the outset of all considerations. The structure equations clearly exhibit the dichotomy in cochlear signal processing.

    Download Mathematical Principles Of Signal Processing: Fourier And Wavelet Analysis

    The signals can either be analyzed in terms of their phase or in terms of their amplitudes. Then the second equation. Inserted in the first equation. The controller processor 8 receives a frame of a certain number of sets of digitally sampled wavelet data from the sampler 5. The controller processor 8 does data compression on this frame to produce a reduced amount of digital data. One way to do data compression is to do a differential encoding in which the difference between the digitally sampled wavelet data from one set in the frame and the digitally sampled wavelet data from the previous set in the frame is stored.

    Another type of data compression discards the digitally sampled wavelet data that is not needed. The controller processor 8 can then store the reduced amount of digital data in the memory This memory 12 can be either external or internal to the chip with the controller processor 8. Later, the controller processor 8 can decompress the reduced amount of digital data to form decompressed digital data.

    This decompressed digital data is sent to circuitry 15 that produces reconstituted analog wavelet outputs. The circuitry 15 comprises an digital-to-analog converter 14 and an analog sample-and-hold The digital-to-analog converter 14 produces an analog signal from the reconstituted digital data. The analog sample-and-hold 16 then samples this analog signal to send reconstituted analog wavelet outputs to the correct input of the wavelet transformer circuitry 2b of the analog wavelet transform chip 2.

    The analog sample-and-hold 16 is under the control of the controller processor 8. The output of the wavelet transformer circuitry 2b is a reconstituted analog input signal produced from the reconstituted analog wavelet outputs. This output of the wavelet transformer circuitry 2b can amplified in amplifier The apparatus shown in FIG. The center frequencies of the filter functions used by the analog wavelet chip 2, shown in FIG.

    Spacing the center frequencies of the filter functions on a logarithmic scale is a convenient way to arrange the information content of speech signals. In the preferred embodiment, the center frequencies of the substantially Gaussian filter functions are at 9 kHz, 4. The bandwidth of each substantially Gaussian filter is set on a logarithmic scale as well. The Gaussian filter functions overlap those in adjacent bands to prevent the overall frequency domain transfer function from having notches.

    Note that the Gaussian filter functions approximately add up to a constant function between the lowest and highest center frequency. If the outputs of the analog wavelet chip are sampled at the points X marked on the frequency-time representation, notice that the sampling of the analog wavelet outputs can be easily time division multiplexed into an sampled output stream. Each "box" in the time frequency plane has an equal area so each sample is of equal value. When the Analog wavelet outputs are sampled at these locations digital wavelet coefficients are formed. In the preferred embodiment, both real and imaginary outputs are sampled.

    The analog wavelet transform chip has three main sections. A modulator function section 20, a wavelet decomposition section 22 and a wavelet reconstruction section The wavelet decomposition section 22 corresponds to the analog wavelet transformer circuitry 2a shown in FIG. The wavelet reconstruction section 24 shown in FIG. In order to simplify the process of designing bandpass filters of adjustable center frequency and width, a complex demodulation process is used.

    The analog input signal is modulated by a signal with a frequency equal to the center frequency of the substantially Gaussian function that is desired to be approximated. In this manner, the filters in the Gaussian filter section 22a may be half-Gaussian shaped filters which are in effect low pass filters. These half-Gaussian shaped filters are easier to design than designing a number of Gaussian shaped filters with different center frequencies. In effect, one design can be used for all the low pass filters in the Gaussian filter section 22a.

    The bandwidth of the low pass filters in the Gaussian filter section 22a can be set by the voltage, V gauss , sent to the low pass filters of the Gaussian filter section 22a. The method of complex demodulation is described below. The input function is designated by f in t. Since the low-pass filter LPF is assumed to be real, and f in t is real, then the output f out t must necessarily be complex, and can be represented by. Looking at FIG.

    Time, Frequency and Fourier

    An oscillator 30 may be external or internal to the analog wavelet chip. This oscillator 30 provides a squarewave signal to a divide-down flip-flop 32 and to divide-down flip-flop The output of the divide down flip-flop 32 and the output of the divide down flip-flop 34 is then sent to the attenuate and bias sections 36 and 38, respectively, and to low pass filters 40 and 42, respectively.

    These low pass filters produce a sinusoidal output out of the squarewave signals from the attenuate and bias circuitry 36 and Due to the complex demodulation process, the real and imaginary parts of the analog wavelet outputs are shifted to a lower frequency. This will not effect the data compression because the frequency bandwidth of the analog wavelet outputs contains the information content of the analog wavelet outputs.

    The frequencies of the analog wavelets are later shifted back in the wavelet reconstruction section 24 as described below. The modulator function section 20 produces sine and cosine modulating pairs for modulating the analog input signal. The output of the divide down flip-flop 32 is sent to another set of flip-flops. In this manner, the frequency of the modulating pairs for the next set of output "slices" is one half that of the previous set of output "slices".

    In general, the use of divide-by-n flip flops, such as flip-flops 26 and 28, allows for the center frequencies of the substantially Gaussian filter functions to be logarithmically spaced. In order to remodulate the signal back to its original frequency, the following multiplication is performed:. This is the same function as the demodulation except for the change in sign. We multiply out the real and imaginary parts of this equation to get a purely real result, which is the exact reconstruction of the original input:. Here the signs have worked out such that the remodulating sinusoidal signals have exactly the same phase relation as the demodulating sinusoidal signals.

    Note that no low-pass filter is needed for reconstruction. The reconstruction of the input signal is done in the wavelet reconstruction section The summation of the signals from multipliers 52 and 54 produce a remodulated signal, f remod6 t. For this reason, looking at FIG. In the preferred environment, there are 12 such "slices" real and imaginary parts of six outputs.

    This "slice" contains the divide down flip-flop 60 that halves the frequency of a signal input to the flip-flop. A follower aggregator 62 is used to bias and attenuate the clock signal from the flip-flop A second order section 64 acts as a low pass filter to produce a sinusoidal wave from the square-wave input. Additionally, a multiplier 66 multiplies input signal, f in t , with the modulating signal from the second order section The substantially half-Gaussian filter 68 filters this multiplied signal to produce the real or imaginary portion of an analog wavelet output.

    A reconstituted analog wavelet output is sent to the multiplier 70 which multiplies this reconstituted analog wavelet output with the modulated signal from the second order section 64 to form a demodulated signal. This demodulated signal is then added to the demodulated signals from the other "slices" in the follower-aggregator 72 to produce a reconstituted analog input signal. In the complex demodulation process, the analog input signal is modulated with a modulating signal of the center frequency of desired passband.

    For this reason, it is only necessary to use a substantially half-Gaussian filter, which is in effect a type of low pass filter. When the signals are remodulated to their respective center frequencies, this half-Gaussian shaped filter will behave as if it were reflected symmetrically across the modulation frequency, becoming a band-pass filter with Gaussian characteristics.

    Time, Frequency and Fourier

    Design of the half-Gaussian shaped filter is based upon a probability argument. The following transfer function describes the filter of the preferred embodiment: EQU1. This filter consists of a cascade of n follower-integrator sections in series. A proof of this equation can be found in Siebert, Circuits, Signals, and Systems, p. The bandwidth of each half-Gaussian filter is automatically set with respect to the others with the exception of the first and last filters, which have bandwidths adjustable using two control voltages. In terms of the transfer function given above for the Gaussian low-pass filter, the parameters T of the highest- and lowest-frequency filters is fixed by these control inputs.

    The sections a, b, c, d and e of the polysilicon strip are equal in resistance so that the voltages V gauss at points , , , , and will have a linear relation. The bandwidth of these substantially half-Gaussian filters then will then vary exponentially in proportion with the value of the center frequency. Thus, the bandwidth for the real and the imaginary part of the highest frequency filter is twice the bandwidth of the real and the imaginary part for the next highest filter and so on.

    Most of the circuits in the wavelet transform chip use transistors that operate in the sub-threshold region i. V gs is the gate source voltage. This circuit is useful in a lot of analog Very Large Scale Integration VLSI situations and is used in the follower-aggregator 62, in the second order section 64, in the half-gaussian filter 68, and in the follower-aggregator 72 all shown in FIG. This circuit is used in two different ways: First, as a current-output device, for which the output function is EQU4 where I bias is the bias current.

    Second, the circuit acts as a voltage-output device, for which the small signal voltage gain is EQU5 which is on the order of to The large-signal output voltage in this case is fixed by bias circuitry at the amplifier output. The total current I bias through both sides of the differential amplifier is fixed by the bias voltage V bias according to the simplified transistor equation: EQU6 this bias voltage is generally set to the range 0. This inverter is the low power analog VLSI equivalent of the latch. The bias voltage V bias is set to place the transistor in the sub-threshold region.

    The switching points of the hysteretic inverter are close to power and ground, and can be adjusted slightly by altering the values of P bias and V bias. The resistor amplifier-follower circuit is used to translate current into voltages in a linear fashion. A transconductance amplifier with a fairly low gain is connected as a follower and thus acts as follower aggregator.

    The low gain of the amplifier increases the region in which the effective resistance is linear. The resistor amplifier-follower is shown as elements 67 and 71 in FIG. These two circuits are digital circuits that produce a digital squarewave output. The circuit of FIG. The divide-down flip-flop circuits of FIGS. The digital sqarewave output is attenuated and biased in the attenuate and bias circuitry 36 and The output current is a multiplicative function of four differential inputs.

    In this case, I bias is computed from the bias voltage V bias in exactly the same manner as for the transconductance amplifier. Although the output undergoes nonlinear tanh compression at extreme values of differential input, the tanh function is approximately linear for small values of the input.

    In the wavelet transform, we wish to multiply two signals together, both referenced to the same midrange 2. V1 - or Vin1 is one input signal, and V2 - or Vin2 is the other input signal. This follower aggregator is used to add the reconstituted real and imaginary analog wavelet outputs to form the reconstituted analog input signal. This circuit is shown as summer 56 in FIG. While each amplifier in the aggregator attempts to follow its own input, the aggregator's output is pulled to an average value of all the amplifier's inputs. The outputs may be tied together because in that arrangement the current outputs from each amplifier adjust to make the total current zero according to Kirchoff's law.

    Each input can be weighted in relation to the others by adjusting the bias voltage on the associated amplifier. For the purposes of the wavelet chip, we always want a pure average, so the bias voltages of all the amplifiers are tied together. When this is done, the equation of the output is no longer a function of the bias voltage which should therefore be set to some reasonable value such as 0.

    The equations relating to the width of the substantially half-Gaussian filter are discussed above. The second order section is used to produce a sinusoidal wave from a square wave input. The second order section 64 is shown in FIG. The purpose of the second-order section is to act as a low-pass filter with a fairly sharp cutoff frequency.

    The input-to-output transfer function of the second-order section is: EQU The analog wavelet chip has pins to set the voltages V. The voltages V. This voltage control oscillator is used to produce the square wave that is sent to the flip-flops used in forming the modulating signals. This circuit actually consists of two oscillators; each oscillator controls the behavior of the other, so that the frequency will drift only half as much as it would for a single oscillator.

    The frequency f of the oscillator is proportional to the exponential bias voltage V 1. Empirical results from simulation give an equation for frequency vs. While the invention has been described by reference to various embodiments, it will be understood that various modifications may be made without departing from the scope of the invention, which is to be limited only by the appended claims. Effective date :