### Table 1: A time-frequency dictionary

2006

### Table 2.1: Some Desirable Properties of Time-Frequency Distributions distributions, they should be real and positive functions. Also, the time marginal of a time- frequency distribution is the integral of the time-frequency distribution over frequency, and the time marginal should be identical to the distribution of signal energy over time. These properties and several others are listed in Table 2.1. For a more complete listing see [35]. 2.5 Covariance and Invariance The notion of a density function being covariant or invariant to an operator plays an important role in time-frequency analysis. Three commonly used operators are the time- shift operator, the frequency-shift operator, and the scale operator de ned, respectively as:

1999

Cited by 11

### TABLE I THE ESTIMATED ERRORS IN THE TIME AND FREQUENCY DOMAINS. Zoomed Filtered

### Table 1: Comparison of the speech recognition performances of STAP features incorporated with various time-frequency pattern activity describing parameters.

2004

"... In PAGE 3: ... These experiments are con- ducted in a hybrid HMM system whose MLP takes 9 frames of contextual input. Table1 gives a comparison of the speech recognition performances when the activation describing pa- rameters incorporated in STAP features is varied. First column in the table gives the description of features used and the second column gives the word error rates.... In PAGE 3: ... Thus in the next set of experiments, the contextual input to the MLP is increased to 19. The results of these experiments are given in the third column of the Table1 . Results show an improvement in recognition performance with more contextual input for the case of STAP feature vectors, where as for MFCC features it is not the case.... In PAGE 4: ...Table 1: Comparison of the speech recognition performances of STAP features incorporated with various time-frequency pattern activity describing parameters. these experiments, the full set of STAP parameters as given in the final row of Table1 is used as the STAP feature. First two rows in the table gives the recognition performances of these features.... In PAGE 4: ... The STAP feature gives a significantly better recogni- tion performance in high noise conditions3. The last row of the Table1 give results of a multi-stream combination of STAP and MFCC features in the tandem frame work. The algorithm used for the combination is as follows: The combination is performed at the posterior outputs of the MLPs corresponding to the two features, based on the entropy values of the MLP outputs[9].... In PAGE 4: ... Using these features as a complementary features, a multi-stream combination of STAP with MFCC show a robust performance in all conditions. These results point to an interesting future work, where a better param- eterization of the activity in the time-frequency pattern, possi- 3The recognition performances of STAPfeatures given in Table1 for noisy conditions are in fact comparable to the performance of standard noise robust features reported in [10, 7]. bly in the neural network frame work could yield better features than the simple parameterization along the time and frequency axes that is used in this paper.... ..."

Cited by 1

### Table 1: CCs for Time and Frequency domain

"... In PAGE 7: ... Values close to 0% indicate that the two variables under test are independent of each other (uncorrelated). Table1 gives the CCs for the three domains at VDD1 and VDD5 for the PLA and ALU transients. VDD5 is cho- sen since it is positioned close to the signal propagation path through the ALU (see Figure 5).... ..."

### Table 5: Biorthogonal Scalets Time-Frequency Uncertainty tfu(a0) and tfu(s0).

1998

Cited by 6

### Table 2.2: Some Desirable Properties and Corresponding Kernel Constraints smooth. A simple, but useful, observation is that by using a kernel that acts as a low pass lter, one can attenuate cross terms and retain the auto terms. This was rst observed by Choi and Williams [15] when they created a Gaussian kernel of the form: ( ; ) = e? 2 2= where the parameter controls the cut o frequency of the lter. This work provided new insight into creating time-frequency distributions and inspired many other methods for creating kernels which we shall not go into here. Further work in this area has shown that many of the desirable properties of time-frequency distributions can be satis ed by placing constraints on the kernel function. In Table 2.2 we list su cient conditions for time- frequency distributions in Cohen apos;s class to satisfy the desirable properties in Table 2.1. All time-frequency distributions in Cohen apos;s class will be covariant to time shifts and frequency shifts. In addition, time-frequency distributions in Cohen apos;s class will also be scale covariant if the kernel satis es:

1999

Cited by 11