and consequently, F'+ = Iy [the (N x N) identity matrix]. We next recall some simple facts. If we view x as a column vector in C% with entries x[0], «[1],...,2[N — 1], the DFT is accomplished by premultiplying with the matrix ee ef PO a, an We are not aware of any straightforward analog of the Heisen- berg inequality for signals in 7{;y. One problem is that the “‘po- sition operator” x(t) ++ tx(t) is well defined on L?(R) but not on Hy, where ¢ takes values in the group D (equipped with addition modulo N). For x : D — C, with ||z||2 = 1 consider the following seemingly obvious analog for the mean: n= ean n|a(n)|?. The immediate problem is that 7 need not belong to the set D. A more serious problem is that addition modulo N, with respect to which the DFT is defined (unlike the continuous Fourier transform), is not the same as addition of rea numbers. In particular, for signals in 7{,,, the mean of the trans- lated signal will only rarely be the translate of the mean. The implied periodicity inherent in both the signal and its DFT are also problematic with respect to translation of the independent variable: The element 0 in D is associated with N, but these two representatives yield different values for the mean Z. dilation operator in the sense of “zooming in and out.” This is because of the inherently discrete (countable) na- ture of the domain. In the continuous case dilation formula Dax(t) = a~@/?)a(a74), the role of the constant a7 1/2) is to maintain ||z|| = ||D.2||. In our digital case, a~' makes sense only when a is inelatively prime to N, and acre (modulo N) by a~! simply permutes {0,1,. — ih. So, [jr|? = Np lela? = SN jalan, and co sequently, we cannot multiply by a~“/2) if we wish to preserve the norm. So, discrete dilation should be defined by Dax{n] = x[a~*n]. However, n + a7'n is a pseudo-random permutation without a physical interpretation, so unlike trans- lations, we choose not to highlight dilations. Fig. 1. N = K? = 25-point optimizer—the picket fence signal. Fig. 4. Spectrum of the 25-point discretized Gaussian pulse. Fig. 5. Improvement of the Hirschman concentration measure for the optimizing pulse over the Gaussian (bottom) pulses. dB versus K = VN. Fig.6. (Top) HOT and (bottom) DFT coefficients for a pure tone of frequency 0.4 normalized. Fig. 7. Squared error versus SNR (DFT is dashed, HOT is dotted, and DCT is the dash-dot). Fig. 8. Squared error versus threshold (DFT is dashed, HOT is dotted, and the DCT is the dash-dot). Fig. 9. Squared error versus the transform length, N = Kk’? (DFT is dashed, HOT is dotted, and the DCT is the dash-dot). Fig. 10. Squared error versus the frequency (DFT is dashed, HOT is dotted, and the DCT is the dash-dot). Dr. DeBrunner has served as an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and he is currently an Associate Editor for both the IEEE SIGNAL PROCESSING LETTERS and the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I. He is also a member of the steering committee for the Asilomar Conference on Signals, Systems, and Computers. We performed a very simple experiment that indicates that the HOT is superior to the DFT and DCT in terms of its ability to separate or resolve two limiting cases of localization in fre- quency, viz. pure tones and additive white noise. We believe that these differences arise from the use of entropy rather than energy as an optimality criterion and are intimately related to the apparent incongruence between the infinitely supported con- tinuous-time case and the finitely supported discrete-time case. It is of great future interest to investigate the dual problem of time-transient estimation in noise and the intermediate case of gated sinusoids in noise.
![and consequently, F'+ = Iy [the (N x N) identity matrix]. We next recall some simple facts. If we view x as a column vector in C% with entries x[0], «[1],...,2[N — 1], the DFT is accomplished by premultiplying with the matrix ee ef PO a, an We are not aware of any straightforward analog of the Heisen- berg inequality for signals in 7{;y. One problem is that the “‘po- sition operator” x(t) ++ tx(t) is well defined on L?(R) but not on Hy, where ¢ takes values in the group D (equipped with addition modulo N). For x : D — C, with ||z||2 = 1 consider the following seemingly obvious analog for the mean: n= ean n|a(n)|?. The immediate problem is that 7 need not belong to the set D. A more serious problem is that addition modulo N, with respect to which the DFT is defined (unlike the continuous Fourier transform), is not the same as addition of rea numbers. In particular, for signals in 7{,,, the mean of the trans- lated signal will only rarely be the translate of the mean. The implied periodicity inherent in both the signal and its DFT are also problematic with respect to translation of the independent variable: The element 0 in D is associated with N, but these two representatives yield different values for the mean Z. dilation operator in the sense of “zooming in and out.” This is because of the inherently discrete (countable) na- ture of the domain. In the continuous case dilation formula Dax(t) = a~@/?)a(a74), the role of the constant a7 1/2) is to maintain ||z|| = ||D.2||. In our digital case, a~' makes sense only when a is inelatively prime to N, and acre (modulo N) by a~! simply permutes {0,1,. — ih. So, [jr|? = Np lela? = SN jalan, and co sequently, we cannot multiply by a~“/2) if we wish to preserve the norm. So, discrete dilation should be defined by Dax{n] = x[a~*n]. However, n + a7'n is a pseudo-random permutation without a physical interpretation, so unlike trans- lations, we choose not to highlight dilations.](https://figures.academia-assets.com/86983343/figure_001.jpg)
