In the present section, we examine a case study demonstrating TF filtering through the complexity-scalable ST. Our ultimate aim is evaluating the improvement in terms of reconstruction accuracies that is achievable by allowing for increasing degrees of redundancy in the TF representation. The synthesis is performed through the two reconstruction procedures based on either inverting the ST or IF estimation, in addition to the octave reverse specific for the minimal sampling scheme. In the scenario under study, the following frequency-modulated waveform is considered
s[n] =exp j(2π(f0+g n)n+φ0)
(2.53) the frequency rate of which is given by
g= 0.37−20/N
2(N−1) (2.54)
and it is shifted in phase by φ0 = π/4, with initial frequency f0 = 20/N. Eq. 2.53 describes a complex-valued linear chirp sampled with a unitary sampling step, and is designed to scan the positive frequencies in the range p ∈ [20/N, 0.37], hence conveniently avoiding the bandwidths heavily affected by IF ambiguity and self- aliasing, both explained before. The signal analyzed is expressed as
x[n] =s[n] +d[n] (2.55) with N = 512; it is the superposition of the waveform given by Eq. 2.53and the following pair of bursts
d[n] = ej2π96nN , n=64, ..., 159 ej2π64nN , n=224, ..., 319 0, otherwise. (2.56)
The task is erasing the bursts smearing the chirp, before retrieving the filtered signal. For this purpose, a TF filter is put in place in the form of binary mask, which is multi- plied point by point with the matrix computed through the forward ST. For the sake of simplicity, the employed mask blanks ideally the TF components containing more
FIGURE2.12: Amplitude the original ST of the signal given by Eq.
2.55. This TF representation coincides with that of the maximal sam- pling scheme for α = exp π2N2/2 and β = N/2, except for the zero-frequency voice, which is undersampled for any combination of
αand β.
FIGURE2.13: Amplitude the FST of Eq. 2.55, which coincides with the TF representation sampled for α=9 and β=1.
energy from d[n]than from s[n]. This simplification is meant to provide a common framework for the energy detection and is not a strong one, since is the comparison between the performance evaluated for STs of different redundancy that matters. The amplitude of the ST together with the respective filter mask is shown in Figs.
2.12-2.14for the fully redundant ST, and the one-to-one FST. The performance for this case study is evaluated in terms of complexity reduction versus reconstruction accuracy. The accuracy is expressed in terms of the normalized root-mean-square
FIGURE2.14: Amplitude of the nearest-neighbor interpolation of the TF representation in Fig.2.13as a comparison with that in Fig.2.12.
FIGURE2.15: Reconstruction accuracy in NRMSE of the filtered sig- nal synthesized through the TI for different pairs of α and β, in com- parison with that obtained by means of the octave reverse, and with
the reference error in dotted line.
error (NRMSE) defined by
NRMSE= ∑ N−1 n=0 |ε[n]|2 ∑N−1 n=0 |y[n]|2 !12 (2.57)
FIGURE2.16: Reconstruction accuracy in NRMSE of the filtered sig- nal directly synthesized through IF estimation using different pairs of αand β, in comparison with that of the indirect estimation from the
peak-interpolation of the N2-point ST.
where y[n]is the signal synthesized from the filtered ST, and ε[n]is the error defined as x[n] −y[n]. Fig. 2.15depicts the reconstruction accuracies achieved with the oc- tave reverse, and the TI, implemented using different combinations of α and β. The reference is the NRMSE calculated between x[n]and s[n]as in Eq. 2.57. We observe that the NRMSEs of the TI are not strictly monotonic functions with respect to β, since there are a few inflections points barely noticeable (i.e., by fractions of deci- bel). Their presence is not meaningful because it is justified by two reasons. First, not every sampling scheme contains the TF components sampled by lower-density schemes, since the frequency domain is sampled nonlinearly. For instance, given constant α and increasing β, the new and larger set of frequency indices does not necessarily include the old and smaller set that was sampled with a higher spectral compression ratio. This fact follows from the nonlinear growth defined in (??) and is evident from Fig.2.9. The second reason is simply due to the error introduced by interpolation. As for the FI, the resultant NRMSEs in Fig.2.15are not visible because they are higher than the reference error for any α, except when α= exp π2N2/2, which returns -10.94 dB. These results confirm that the FI is only suitable to retrieve the filtered signal from the fully redundant ST, and is not capable to cope neither with time decimation nor spectral compression. Alternatively, the synthesis can be carried out through IF estimation. The results in Fig. 2.16are obtained using Eq.
2.29to directly estimate the amplitude and the phase of the filtered signal. Instead, the indirect reconstruction based on IF peak-interpolation estimation was found to perform well only without compression; therefore, for the sake of comparison, the respective NRMSE is calculated only for the original N2-point ST. An important ob- servation can be made from Fig.2.16that is increasing β above a certain value does
not bring significant improvement to the signal synthesized through IF estimation. Overall, the results suggest that the applications that employ the ST to modify the analyzed signal TF characteristics can benefit in terms of accuracy from a flexible TF sampling scheme. In fact, a level of redundancy aids the discrimination of the TF components to be filtered, and it also lessens the artifacts inevitably distorting the information due the filter mask itself. For example, Fig.2.15shows that the NRMSE falls by more than 3 dB with respect to that obtained using the FST when β is in- creased from 1 to 2, while the size of the corresponding ST is still less than 2N, as clear from Fig. 2.10. Finally, the pair of parameters α and β can be tuned to rescale the TF representation according to a controllable trade-off between complexity and reconstruction accuracy.