Patent Publication Number: US-9842611-B2

Title: Estimating pitch using peak-to-peak distances

Description:
PRIORITY CLAIM 
     This application claims priority to U.S. Provisional Patent Application No. 62/112,832, entitled “PEAK INTERVAL PITCH ESTIMATION,” filed Feb. 6, 2015, the entirety of which is incorporated herein by reference. 
    
    
     BACKGROUND 
     A harmonic signal may have a fundamental frequency and one or more overtones. Harmonic signals include, for example, speech and music. A harmonic signal may have a fundamental frequency, which may be referred to as the first harmonic. A harmonic signal may include other harmonics that may occur at multiples of the first harmonic. For example, if the fundamental frequency is f at a certain time, then the other harmonics may have frequencies of 2f, 3f, and so forth. 
     The fundamental frequency of a harmonic signal may change over time. For example, when a person is speaking, the fundamental frequency of the speech may increase at the end of a question. A change in the frequency of a signal may be referred to as a chirp rate. The chirp rate of a harmonic signal may be different for different harmonics. For example, if the first harmonic has a chirp rate of c, then other the harmonics may have chirp rates of 2c, 3c, and so forth. 
     In applications, such as speech recognition, signal reconstruction, and speaker recognition, it may be desirable to determine properties of a harmonic signal over time. For example, it may be desirable to determine a pitch of the signal, a rate of change of the pitch over time, or the frequency, chirp rate, or amplitude of different harmonics. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
       The invention and the following detailed description of certain embodiments thereof may be understood by reference to the following figures: 
         FIG. 1  illustrates examples of harmonic signals with different fractional chirp rates. 
         FIG. 2  illustrates a spectrogram of a portion of a speech signal. 
         FIG. 3  illustrates a representation of a harmonic signal over frequency and chirp rate. 
         FIG. 4  illustrates a representation of a harmonic signal over frequency and fractional chirp rate. 
         FIG. 5  illustrates two examples of a generalized spectrum of a signal. 
         FIG. 6  illustrates a pitch velocity transform of a speech signal. 
         FIG. 7  illustrates two examples of generalized spectra of a speech signal. 
         FIG. 8  illustrates an LLR spectrum of a speech signal. 
         FIG. 9A  illustrates peak-to-peak distances for a single threshold in an LLR spectrum of a speech signal. 
         FIG. 9B  illustrates peak-to-peak distances for multiple thresholds in an LLR spectrum of a speech signal. 
         FIG. 10A  illustrates frequency portions of a frequency representation of a speech signal for a first pitch estimate. 
         FIG. 10B  illustrates frequency portions of a frequency representation of a speech signal for a second pitch estimate. 
         FIG. 11  is a flowchart showing an example implementation of computing features from a signal. 
         FIG. 12  is a flowchart showing an example implementation of estimating a fractional chirp rate from a signal. 
         FIG. 13  is a flowchart showing an example implementation of estimating a pitch of a signal using peak-to-peak distances. 
         FIG. 14  is a flowchart showing an example implementation of estimating a pitch of a signal using correlations. 
         FIG. 15  is an exemplary computing device that may be used to estimate features of signals. 
     
    
    
     DETAILED DESCRIPTION 
     Described herein are techniques for determining properties of a harmonic signal over time. For example, the properties of a harmonic signal may be determined at regular intervals, such as every 10 milliseconds. These properties may be used for processing speech or other signals, for example, as features for performing automatic speech recognition or speaker verification or identification. These properties may also be used to perform a signal reconstruction to reduce the noise level of the harmonic signal. 
     The relationship between the harmonics of a harmonic signal may be used to improve the estimation of the properties of the harmonic signal. For example, if the first harmonic of a harmonic signal has a frequency of f and a chirp rate of c, then it is expected that the higher harmonics have frequencies at multiples of f and chirp rates at multiples of c. Techniques that take advantage of these relationships may provide better results than other techniques. 
     A harmonic signal may have a pitch. For some harmonic signals, the pitch may correspond to the frequency of the first harmonic. For some harmonic signals, the first harmonic may not be present or not visible (e.g., may be covered by noise), and the pitch may be determined from a frequency difference between the second and third harmonics. For some harmonic signals, multiple harmonics may be present or not visible, and the pitch may be determined from the frequencies of the visible harmonics. 
     The pitch of a harmonic signal may change over time. For example, the pitch of a voice or the note of a musical instrument may change over time. As the pitch of a harmonic signal changes, each of the harmonics will have a chirp rate, and the chirp rate of each harmonic may be different. The rate of change of the pitch may be referred to as pitch velocity or described by a fractional chirp rate. In some implementations, the fractional chirp rate may by computed as x=c n /f n  where χ represents the fractional chirp rate, c n  represents the chirp rate of the nth harmonic, and f n  represents the frequency of the n th  harmonic. 
     In some implementations, it may be desired to compute the pitch and/or fractional chirp rate of a harmonic signal at regular intervals. For example, it may be desired to compute the pitch and/or fractional chirp rate every 10 milliseconds by performing computations on a portion of the signal that may be obtained by applying a window (e.g., a Gaussian, Hamming, or Hann window) to the signal. Successive portions of the signal may be referred to as frames, and frames may overlap. For example, frames may be created every 10 milliseconds and each frame may be 50 milliseconds long. 
       FIG. 1  illustrates examples of four harmonic signals with different fractional chirp rates as a function of time and frequency.  FIG. 1  does not represent actual signals but provides a conceptual illustration of how chirplets (Gaussian signals with a specified time, frequency, chirp rate, and duration) would appear in a time-frequency representation, such as a spectrogram. 
     Harmonic signal  110  is centered at a time t 1  and has four harmonics. The first harmonic has a frequency of f and the second, third, and fourth harmonics have frequencies of 2f 3f and 4f respectively. Each of the harmonics has a chirp rate of 0 since the frequency of the harmonics is not changing over time. Accordingly, the fractional chirp rate of harmonic signal  110  is 0. 
     Harmonic signal  120  is centered at time t 2  and has four harmonics. The first harmonic has a frequency of 2f and the second, third, and fourth harmonics have frequencies of 4f 6f and 8f respectively. The first harmonic has a chirp rate of c that is positive since the frequency is increasing over time. The second, third, and fourth harmonics have chirp rates of 2c, 3c, and 4c, respectively. Accordingly, the fractional chirp rate of harmonic signal  120  is c/2f. 
     Harmonic signal  130  is centered at time t 3  and has four harmonics. The first harmonic has a frequency of f and the second, third, and fourth harmonics have frequencies of 2f 3f and 4f respectively. The first harmonic also has a chirp rate of c, and the second, third, and fourth harmonics have chirp rates of 2c, 3c, and 4c, respectively. Accordingly, the fractional chirp rate of harmonic signal  130  is c/f, which is twice that of harmonic signal  120 . 
     Harmonic signal  140  is centered at time t 4  and has four harmonics. The first harmonic has a frequency of f and the second, third, and fourth harmonics have frequencies of 2f, 3f and 4f respectively. The first harmonic has a chirp rate of 2c as the rate of change of frequency is double that of harmonic signal  130 . The second, third, and fourth harmonics have chirp rates of 4c, 6c, and 8c, respectively. Accordingly, the fractional chirp rate of harmonic signal  140  is 2c/f, which is twice that of harmonic signal  130 . 
       FIG. 2  illustrates a spectrogram of a portion of a speech signal. In the spectrogram, multiple harmonics are visible. At each time instant in the spectrogram, the harmonics have the relationship described above. For example, at each time instant, the frequency and chirp rate of the second harmonic is about twice the frequency and chirp rate of the first harmonic. 
       FIG. 3  illustrates examples of four harmonic signals as a function of frequency and chirp rate, which will be referred to herein as a frequency-chirp distribution or representation.  FIG. 3  does not represent actual signals but provides a conceptual illustration of how the harmonic signals of  FIG. 1  would appear in a representation of frequency and chirp rate. In computing a frequency-chirp representation, there may not be a time variable so the frequency-chirp distribution may represent an entire signal and not a portion of the signal at a particular time. In some implementations, it may be desired to compute a frequency-chirp distribution for portions of a signal corresponding to different times. For example, it may be desired to compute a frequency-chirp distribution every 10 milliseconds, by applying a sliding window to the signal. 
       FIG. 3  may be constructed conceptually by reviewing the frequency and chirp rate of the harmonics of the harmonic signals of  FIG. 1 . For example, for harmonic signal  110 , each of the chirp rates is 0, and the frequencies of the four harmonics are 2f, 3f, and 4f, respectively. Accordingly, the four harmonics of harmonic signal  110  are represented in these locations in  FIG. 3 . Similarly, the harmonics of harmonic signals  120 ,  130 , and  140  are represented in  FIG. 3  according to their respective frequencies and chirp rates from  FIG. 1 . 
     A frequency-chirp distribution may be computed using techniques similar to computing a time-frequency distribution, such as a spectrogram. For example, in some implementations, a frequency-chirp distribution may be computed using an inner product. Let FC(f,c) represent a frequency-chirp distribution where f corresponds to a frequency variable and c corresponds to a chirp rate variable. A frequency-chirp rate distribution may be computed using inner products as
 
 FC ( f,c )=   x ,ψ( f,c ) 
 
where x is the signal being processed (or a windowed portion of it) and ψ(f,c) is a function parameterized by frequency f and chirp rate c. In some implementations, ψ(f,c) may represent a chirplet, such as
 
               ψ   ⁡     (     f   ,   c     )       =       1       2   ⁢   π   ⁢           ⁢     σ   2           ⁢     e         -     1   2       ⁢       (       t   -     t   0       σ     )     2       +       f   ⁡     (     t   -     t   0       )       ⁢   i     +       c   2     ⁢       (     t   -     t   0       )     2     ⁢   i                 
where σ corresponds to a duration or spread of the chirplet and t 0  is a location of the chirplet in time. To compute a distribution of frequency and chirp rate, one can select an appropriate function ψ(f,c), such as a chirplet, and compute FC(f,c) for multiple values of f and c. A frequency-chirp distribution is not limited to the above example, and may be computed in other ways. For example, a frequency-chirp distribution may be computed as the real part, imaginary part, magnitude, or magnitude squared of an inner product, may be computed using measures of similarity other than an inner product, or may be computed using non-linear functions of the signal.
 
     The four harmonic signals in  FIG. 3  each have different fractional chirp rates. Harmonic signal  110  has a fractional chirp rate of 0, harmonic signal  120  has a fractional chirp rate of c/2f, harmonic signal  130  has a fractional chirp rate of c/f, and harmonic signal  120  has a fractional chirp rate of 2c/f. The dashed and dotted lines in  FIG. 3  thus indicate lines of constant fractional chirp rate. A harmonic centered on the dash-dotted line will have a fractional chirp rate of c/2f, a harmonic centered on the dotted line will have a fractional chirp rate of c/f, and a harmonic centered on the dashed line will have a fractional chirp rate of 2c/f. 
     Accordingly, any radial line in  FIG. 3  corresponds to a constant fractional chirp rate. From this observation one can generate a distribution of frequency and fractional chirp rate, which may be referred to as a pitch-velocity transform (PVT) or a chirprum. A PVT may be denoted as P(f,χ), where f corresponds to a frequency variable and χ corresponds to a fractional chirp rate variable. Conceptually, one may construct a PVT by warping a frequency-chirp distribution so that the radial lines of a frequency-chirp distribution become horizontal lines of a PVT.  FIG. 4  shows a conceptual example of a PVT created from the frequency-chirp distribution of  FIG. 3 . Because each harmonic of a harmonic signal has the same fractional chirp rate, they are aligned horizontally as shown in  FIG. 4 . 
     In some implementations, a PVT may be computed from a frequency-chirp distribution. For example, a PVT may be computed as
 
 P ( f ,χ)= FC ( f,χf )
 
since c=χf as described above. The PVT need not, however, be computed from a frequency-chirp distribution.
 
     A PVT may also be computed using techniques similar to computing a time-frequency distribution, such as a spectrogram. For example, in some implementations a PVT may be computed using an inner product. A frequency-chirp rate distribution may be computed as
 
 P ( f ,χ)=   x ,ψ( f,χf ) 
 
where ψ( ) is a function as described above. To compute a PVT, one can select an appropriate function ψ( ), such as a chirplet, and compute P(f,χ) for multiple values of f and χ. A PVT is not limited to the above example, and a PVT may be computed in other ways. For example, a PVT may be computed as the real part, imaginary part, magnitude, or magnitude squared of an inner product, may be computed using measures of similarity other than an inner product, or may be computed using non-linear functions of the signal.
 
     The PVT for a specified value of a fractional chirp rate is a function of frequency and may be considered to be a spectrum or a generalized spectrum of the signal. Accordingly, for each value of a fractional chirp rate, a generalized spectrum may be determined from the PVT that is associated with a particular fractional chirp rate. The generalized spectra may be referred to as X χ (f). As described below, these generalized spectra need not be computed from a PVT and may be computed in other ways. The PVT for a specified fractional chirp rate corresponds to a slice of the PVT, which will be referred to herein as a row of the PVT (if the PVT was presented in a different orientation, this could also be referred to as a column and the orientation of the PVT is not a limiting feature of the techniques described herein). For clarity of explanation, a chirplet will be used for the function χ( ) in the following discussion, but any appropriate function may be used for ψ( ). 
     For a fractional chirp rate of 0, the PVT corresponds to
 
 P ( f, 0)=   x ,ψ( f, 0) 
 
which corresponds to an inner product of the signal with a Gaussian where the Gaussian has a chirp rate of zero and is modulated to the corresponding frequency f of the PVT. This may be the same as computing a short-time Fourier transform of the signal with a Gaussian window.
 
     For a non-zero fractional chirp rate, the PVT corresponds to an inner product of the signal with a Gaussian where the chirp rate of the Gaussian increases as the frequency of the Gaussian increases. In particular, the chirp rate may be the product of the fractional chirp rate and the frequency. For non-zero chirp rates, the PVT may have an effect similar to slowing down or reducing the fractional chirp rate of the signal (or conversely, speeding up or increasing the fractional chirp rate of the signal). Accordingly, each row of the PVT corresponds to a generalized spectrum where the fractional chirp rate of the signal has been modified by a value corresponding to the row of the PVT. 
     When the fractional chirp rate of the generalized spectrum (or row of the PVT) is equal to the fractional chirp rate of the signal, the generalized spectrum may correspond to removing the fractional chirp rate of the signal and the generalized spectrum for this value of the fractional chirp rate may be referred to as a stationary spectrum of the signal or a best row of the PVT. 
       FIG. 5  illustrates hypothetical generalized spectra (or rows of the PVT) generated using two different values of fractional chirp rate for the harmonic signal  140  of  FIG. 1 . The four peaks ( 511 ,  512 ,  513 ,  514 ) illustrate the generalized spectrum where the fractional chirp rate matches the fractional chirp rate of the signal, and this may be referred to as a stationary spectrum. Because the fractional chirp rate of the row of the generalized spectrum matches the fractional chirp rate of the signal (i) the width of the four peaks may be narrower than the generalized spectra for other fractional chirp rate values, and (ii) the height of the four peaks may be higher than the generalized spectra for other fractional chirp rate values. Because the peaks may be narrower and higher they may be easier to detect than for other generalized spectra. The peaks of stationary spectrum may be narrower and higher because the stationary spectrum may have the effect of removing the fractional chirp rate of the signal. 
     The four peaks ( 521 ,  522 ,  523 ,  524 ) illustrate a generalized spectrum for a fractional chirp rate that is different from the fractional chirp rate of the signal. Because the fractional chirp rate of the generalized spectrum does not match the signal, the peaks may be shorter and wider. 
       FIG. 6  illustrates a PVT of the signal from  FIG. 2  at approximately 0.21 seconds. At this time, the signal has a pitch of approximately 230 Hz and a fractional chirp rate of approximately 4. The PVT shows features of the signal for each of the harmonics. For example, the PVT shows the first harmonic at approximately 230 Hz on the frequency axis and 4 on the fractional chirp rate axis. Similarly, the PVT shows the second harmonic at approximately 460 Hz on the frequency axis and 4 on the fractional chirp rate axis, and so forth. At frequencies between the harmonics, the PVT has lower values because the signal energy is lower in these regions. At fractional chirp rates different from 4, the PVT has lower values because the fractional chirp rate of the PVT does not match the fractional chirp rate of the signal. 
       FIG. 7  illustrates two generalized spectra corresponding to rows of the PVT of  FIG. 6 . The solid line corresponds to a generalized spectrum where the fractional chirp rate matches the fractional chirp rate of the signal (a fractional chirp rate of about 4) or the stationary spectrum. The dashed line corresponds to a generalized spectrum with a fractional chirp of zero, which will be referred to as the zero generalized spectrum (and may correspond to a short-time Fourier transform of the signal). The peaks of the stationary spectrum are higher and narrower than the peaks of the zero generalized spectrum. For the first harmonic, the peak  711  of the stationary spectrum is about twice the height and one-third the width of peak  721  of the zero generalized spectrum. For the third harmonic, the difference between the peak  712  of the stationary spectrum and peak  722  of the zero generalized spectrum is even greater. For the seventh harmonic, the peak  713  of the stationary spectrum is clearly visible, but the peak of the zero generalized spectrum is not visible. 
     The features of different generalized spectra (or rows of the PVT) may be used to determine a fractional chirp rate of the signal. As noted above, the peaks of the generalized spectrum may be narrower and higher for the correct value of the fractional chirp rate. Techniques for measuring narrower and higher peaks of a signal may thus be used for estimating the fractional chirp rate of a signal. 
     To estimate fractional chirp rate, a function may be used that takes a vector (e.g., a spectrum) as input and outputs one or more scores according to some criteria. Let g( ) be a function that takes a vector as input (such as a generalized spectrum or row of a PVT) and outputs a one or more values or scores corresponding to the input. In some implementations, the output of g( ) may be a number that indicates a peakiness of the input. For example, g( ) may correspond to entropy, Fisher information, Kullback-Leibler divergence, or a magnitude of the input to a fourth or higher power. Using the function g( ) the fractional chirp rate of a signal may be estimated from the PVT using the following: 
               χ   ^     =         arg   ⁢           ⁢   max     χ     ⁢           ⁢     g   ⁡     (     P   ⁡     (     f   ,   χ     )       )               
where {circumflex over (χ)} is an estimate of the fractional chirp rate. The function g( ) may be computed for multiple rows of the PVT, and the row producing the highest value of g( ) may be selected as corresponding to an estimated fractional chirp rate of the signal.
 
     The estimate of the fractional chirp rate may also be computed from a frequency chirp distribution, such as the frequency chirp distribution described above: 
     
       
         
           
             
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     The estimate of the fractional chirp rate may also be computed from a generalized spectrum: 
     
       
         
           
             
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                   arg 
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     The estimate of the fractional chirp rate may also be computed using inner products of the signal with the function ψ( ): 
     
       
         
           
             
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     As described above, each of the PVT, the frequency chirp rate distribution, and the generalized spectrum may be computed using a variety of techniques. In some implementations, these quantities may be determined by computing an inner product of a signal with a chirplet, but the techniques described herein are not limited to that particular implementation. For example, functions other than chirplets may be used and measures of similarity other than an inner product may be used. 
     In some implementations, a generalized spectrum may be modified before being used to determine the fractional chirp rate of the signal. For example, a log likelihood ratio (LLR) spectrum may be computed from the generalized spectrum, and the LLR spectrum may be denoted as LLR χ (f). An LLR spectrum may use hypothesis testing techniques to improve a determination of whether a harmonic is present at a frequency of a spectrum. For example, to determine whether a harmonic is present at the frequencies of the stationary spectrum shown in  FIG. 7 , one could compare the value of the spectrum to a threshold. Using an LLR spectrum may improve this determination. 
     An LLR spectrum may be computed using a log likelihood ratio of two hypotheses: (1) a harmonic is present at a frequency of the signal, and (2) a harmonic is not present at a frequency of the signal. For each of the two hypotheses, a likelihood may be computed. The two likelihoods may be compared to determine whether a harmonic is present, such as by computing a ratio of the logs of the two likelihoods. 
     In some implementations, the log likelihood for a harmonic being present at a frequency of the signal may be computed by fitting a Gaussian to the signal spectrum at the frequency and then computing a residual sum of squares between the Gaussian and the signal. To fit a Gaussian to a spectrum at a frequency, the Gaussian may be centered at the frequency, and then an amplitude of the Gaussian may be computed using any suitable techniques for estimating these parameters. In some implementations, a spread in frequency or duration of the Gaussian may match a window used to compute signal spectrum or the spread of the Gaussian may also be determined during the fitting process. For example, when fitting a Gaussian to peak  711  of the stationary spectrum in  FIG. 7 , the amplitude of the Gaussian may be approximately 0.12 and the duration of the Gaussian may correspond approximately to the duration of the peak (or the window used to compute the spectrum). The log likelihood may then be computed by computing a residual sum of squares between the Gaussian and the signal spectrum in a window around the frequency for which the likelihood is being computed. 
     In some implementations, the log likelihood for a harmonic not being present at a frequency may correspond to computing a residual sum of squares between a zero spectrum (a spectrum that is zero at all frequencies) and the signal spectrum in a window around the frequency for which the likelihood is being computed. 
     The LLR spectrum may be determined by computing the two likelihoods for each frequency of the signal spectrum (such as a generalized spectrum) and then computing a logarithm (e.g., natural logarithm) of the ratio of the two likelihoods. Other steps may be performed as well, such as estimating a noise variance in the signal and using the estimated noise variance to normalize the log likelihoods. In some implementations, an LLR spectrum for a frequency f may be computed as 
               LLR   ⁡     (   f   )       =       1     2   ⁢     σ   noise   2         ⁢     (         X   h     ⁢   X     -         (     X   -       G   ^     f       )     h     ⁢     (     X   -       G   ^     f       )         )             
where σ noise   2  is an estimated noise variance, X is a spectrum, h is a Hermitian transpose, and Ĝ f  is a best fitting Gaussian to the spectrum at frequency f.
 
       FIG. 8  illustrates an example of an LLR spectrum corresponding to the stationary spectrum of  FIG. 7 . For each frequency, the LLR spectrum has a high value where a harmonic is present and a low value when a harmonic is not present. The LLR spectrum may provide a better determination than other spectra (such as a generalized or stationary spectra) of whether harmonics are present at different frequencies. 
     The estimate of the fractional chirp rate may also be computed using the LLR spectrum: 
     
       
         
           
             
               χ 
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             = 
             
               
                 
                   arg 
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                   max 
                 
                 χ 
               
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     To illustrate some possible implementations of estimating fractional chirp rate, examples of the function g( ) will be provided. The examples below will use the generalized spectrum, but other spectra, such as the LLR spectrum may be used as well. 
     In some implementations, the fractional chirp rate may be estimated using a magnitude to the fourth power of the generalized spectrum:
 
 g ( X   χ ( f ))=∫| X   χ ( f )| 4   df  
 
     In some implementations, the function g( ) may comprise at least some of the following sequence of operations: (1) compute |X χ (f)| 2  (may be normalized by dividing by the total energy of the signal or some other normalization value); (2) compute an auto-correlation of |X χ (f)| 2  denoted as r X (τ); and (3) compute the Fisher information, entropy, Kullback-Leibler divergence, sum of squared (or magnitude squared) values of r X (τ), or a sum of squared second derivatives of r X (τ). The foregoing examples are not limiting and other variations are possible. For example, in step (1), X χ (f) or its magnitude, or real or imaginary parts may be used in place of |X χ (f)| 2 . 
     Accordingly, the fractional chirp rate of a signal may be determined using any combinations of the above techniques or any similar techniques known to one of skill in the art. 
     In addition to estimating a fractional chirp rate of the signal, a pitch of the signal may also be estimated. In some implementations, the fractional chirp rate may be estimated first, and the estimated fractional chirp rate may be used in estimating the pitch. For example, after estimating the fractional chirp rate, denoted as 2, the generalized spectrum corresponding to the estimated fractional chirp rate may be used to estimate a pitch. 
     When estimating pitch, it is possible that the pitch estimate may be different from the true pitch by an octave, which may be referred to as an octave error. For example, if the true pitch is 300 Hz, the pitch estimate may be 150 Hz or 600 Hz. To avoid octave errors, a two-step approach may be used to estimate pitch. First, a coarse pitch estimate may be determined to obtain an estimate that may be less accurate but less susceptible to octave errors, and second, a precise pitch estimate may be used to refine the coarse pitch estimate. 
     A coarse pitch estimate may be determined by computing peak-to-peak distances of a spectrum, such as a generalized spectrum or an LLR spectrum (corresponding to the estimate of the fractional chirp rate). For clarity in the following explanation, the LLR spectrum will be used as an example spectrum, but the techniques described herein are not limited to the LLR spectrum and any appropriate spectrum may be used. 
     When computing peak-to-peak distances in a spectrum, it may not always be clear which peaks correspond to the signal and which peaks correspond to noise. Including too many peaks that correspond to noise or excluding too many peaks that correspond to signal may reduce the accuracy of the coarse pitch estimate. Although the example LLR spectrum in  FIG. 8  has low noise, for signals with higher noise levels, additional peaks caused by noise may also be present. 
     In some implementations, peaks may be selected from the LLR spectrum using thresholds. For example, a standard deviation (or variance) of the noise in the spectrum may be determined and a threshold may be computed or selected using the standard deviation of the noise, such as setting the threshold to a multiple or fraction of the standard deviation (e.g., set a threshold to twice the standard deviation of the noise). After choosing a threshold, peak-to-peak distances may be determined. For example,  FIG. 9A  shows peak-to-peak distances for a threshold of approximately 0.3. At this threshold, the first 5 peak-to-peak distances are about 230 Hz, the sixth is about 460 Hz, the seventh and eighth are about 230 Hz, and the ninth is about 690 Hz. After determining the peak-to-peak distances, a most frequently occurring peak-to-peak distance may be selected as the coarse pitch estimate. For example, a histogram may be computed with bins with a width of 2-5 Hz, and the histogram bin with the largest number of counts may be selected as the coarse pitch estimate. 
     In some implementations, multiple thresholds may be used as illustrated in  FIG. 9B . For example, thresholds may be selected using the heights of the peaks in the LLR spectrum, such as the ten highest peaks or all peaks above a second threshold (e.g., above twice the standard deviation of the noise). Peak-to-peak distances may be computed for each of the thresholds. In  FIG. 9B , peak-to-peak distance  901  is determined using the tallest peak as a threshold, peak-to-peak distances  911  and  912  are determined using the second tallest peak as a threshold, peak-to-peak distances  921 ,  922 , and  923  are determined using the third tallest peak as a threshold, and so forth. As above, a most frequently occurring peak-to-peak distance may be selected as the coarse pitch estimate, for example, by using a histogram. 
     In some implementations, peak-to-peak distances may be computed for multiple time frames for determining a coarse pitch estimate. For example, to determine a coarse pitch estimate for a particular frame, peak-to-peak distances may be computed for the current frame, five previous frames, and five subsequent frames. The peak-to-peak distances for all of the frames may be pooled together in determining a coarse pitch estimate, such as computing a histogram for all of the peak-to-peak distances. 
     In some implementations, peak-to-peak distances may by computed using different smoothing kernels on the spectrum. Applying a smoothing kernel to a spectrum may reduce peaks caused by noise but may also reduce peaks caused by signal. For noisy signals, a wider kernel may perform better and, for less noisy signals, a narrower kernel may perform better. It may not be known how to select an appropriate kernel width, and thus peak-to-peak distances may be computed from a spectrum for each of a specified group of kernel widths. As above, the peak-to-peak distances for all of the smoothing kernels may be pooled together in determining a coarse pitch estimate. 
     Accordingly, peak-to-peak distances may be computed in a variety of ways including, but not limited to, different thresholds, different time instances (e.g., frames), and different smoothing kernels. From these peak-to-peak distances, a coarse pitch estimate may be determined. In some implementations, a coarse pitch estimate may be determined as the frequency corresponding to the mode of the histogram for all computed peak-to-peak distances. 
     In some implementations, a coarse pitch estimate may be determined by estimating a cumulative distribution function (CDF) and/or a probability density function (PDF) of the peak-to-peak distances instead of using a histogram. For example, a CDF for pitch may be estimated as follows. For any pitch values smaller than the smallest peak-to-peak distance, the CDF will be zero and for any pitch values larger than the largest peak-to-peak distance, the CDF will be one. For a pitch value in between these two bounds, the CDF may be estimated as the cumulative number of peak-to-peak distances smaller than the pitch value divided by the total number of peak-to-peak distances. For example, consider the peak-to-peak distances illustrated in  FIG. 9A .  FIG. 9A  shows a total of 9 peak-to-peak distances, including 7 peak-to-peak distances of 230 Hz, 1 peak-to-peak distance of 460 Hz, and 1 peak-to-peak distance of 690 Hz. A CDF may be estimated as having a value of 0 for frequencies less that 230 Hz, a value of 7/9 for frequencies between 230 Hz and 460 Hz, a value 8/9 for frequencies between 460 Hz and 690 Hz, and a value of 1 for frequencies above 690 Hz. 
     This estimated CDF may resemble a step function, and accordingly the CDF may be smoothed using any appropriate smoothing technique, such as spline interpolation, low-pass filtering, or LOWESS smoothing. The coarse pitch estimate may be determined as the pitch value corresponding to the largest slope of the CDF. 
     In some implementations, a PDF may be estimated from the CDF by computing a derivative of the CDF and any appropriate techniques may be used for computing the derivative. The coarse pitch estimate may then be determined as the pitch value corresponding to the peak of the PDF. 
     In some implementations, multiple preliminary coarse pitch estimates may be determined, and an actual coarse pitch estimate may be determined using the preliminary pitch estimates. For example, an average of the preliminary coarse pitch estimates or a most common coarse pitch estimate may be selected as the actual coarse pitch estimate. For example, a coarse pitch estimate may be computed for each of a group of threshold values. For high threshold values, the coarse pitch estimate may be too high, and for low threshold values, the coarse pitch estimate may be too low. For thresholds in between, the coarse pitch estimate may be more accurate. To determine an actual coarse pitch estimate, a histogram may be computed of the multiple preliminary coarse pitch estimates, and the actual coarse pitch estimate may correspond to the frequency of the mode of the histogram. In some implementations, outliers may be removed from the histogram to improve the actual coarse pitch estimate. 
     After obtaining a coarse pitch estimate, a precise pitch estimate may be obtained using the coarse pitch estimate as a starting point. A precise pitch estimate may be determined using the shape of each harmonic in a spectrum (again, any appropriate spectrum may be used, such as a generalized spectrum, a stationary spectrum, or an LLR spectrum). To compare the shapes of harmonics in the spectrum, portions of the spectrum may be extracted as shown in  FIG. 10A  and  FIG. 10B . 
       FIG. 10A  illustrates portions of a spectrum for a first pitch estimate, where the pitch estimate is very close to the true pitch of the signal. Suppose that the true pitch of the signal is about 230 Hz and the pitch estimate is also about 230 Hz. A portion of the spectrum for each harmonic can be identified by using a multiple of the estimated pitch. In  FIG. 10A , the portion  1010  is at approximately 230 Hz, the portion  1011  is at approximately 460 Hz, and portions  1012 - 1017  are each at higher multiples of 230 Hz. Because the pitch estimate is accurate, each harmonic is approximately centered in the middle of each portion. Examples of estimating pitch in audio signals based on symmetry characteristics are described in U.S. patent application Ser. No. 14/502,844, filed on Sep. 30, 2014 and entitled “SYSTEMS AND METHODS FOR ESTIMATING PITCH IN AUDIO SIGNALS BASED ON SYMMETRY CHARACTERISTICS INDEPENDENT OF HARMONIC AMPLITUDES,” which is incorporated herein by reference in its entirety. 
       FIG. 10B  illustrates portions of a spectrum for a second pitch estimate, where the pitch estimate is slightly lower than the true pitch of the signal. For example, the pitch estimate may be 228 Hz and the actual pitch may be 230 Hz. Again, a portion of the spectrum for each harmonic can be identified using multiples of the pitch estimate. For each harmonic, the portion is slightly to the left of the true position of the harmonic and the offset increases as the harmonic number increases. Portion  1020  is about 2 Hz to the left of the true position of the first harmonic, portion  1021  is about 4 Hz to the left of the true position of the second harmonic, and portions  1022 - 1027  are each increasingly further to the left as the harmonic number increases. For example, portion  1027  is about 16 Hz to the left of the true position of the eighth harmonic. 
     The frequency portions from  FIGS. 10A and 10B  can be used to determine the accuracy of a pitch estimate. When the pitch estimate is accurate, as in  FIG. 10A , each harmonic is centered in a frequency portion and thus the frequency portions all have similar shapes. When the pitch estimate is not accurate, as in  FIG. 10B , each harmonic is off center in the frequency portion and is more off center as the harmonic number increases. Thus, when the pitch estimate is less accurate, the frequency portions are less similar to one another. 
     In addition to comparing the shape of a first frequency portion with a second frequency portion, a frequency portion may be compared to a reversed version of itself since the shape of a harmonic is generally symmetric. For an accurate pitch estimate, a harmonic will be centered in a frequency portion, and thus reversing the portion will provide a similar shape. For an inaccurate pitch estimate, the harmonic will not be centered in the frequency portion, and reversing the portion will result in a different shape. Similarly, a first frequency portion can be compared to a reversed version of a second frequency portion. 
     The frequency portions may have any appropriate width. In some implementations, the frequency portions may partition the spectrum, may overlap adjacent portions, or may have gaps between them (as shown in  FIGS. 10A and 10B ). The frequency portions used may correspond to any frequency representation, such as a spectrum of a signal or a real part, imaginary part, magnitude, or magnitude squared of a spectrum of a signal. The frequency portions may also be normalized to remove differences that are less relevant to determining pitch. For example, for each frequency portion a mean and a standard deviation may be determined, and the frequency portion may be normalized by subtracting the mean value and then dividing by the standard deviation (e.g., a z-score). 
     Correlations may be used to measure whether two frequency portions have similar shapes and to determine if a harmonic is centered at the expected frequency. The frequency portions for a pitch estimate may be determined as described above, and a correlation may be performed by computing an inner product of two frequency portions. Correlations that may be performed include the following: a correlation of a first frequency portion with a second frequency portion, a correlation of a first frequency portion with a reversed version of itself, and a correlation of a first frequency portion with a reversed version of a second frequency portion. 
     The correlations may have higher values for more accurate pitch estimates and lower values for less accurate pitch estimates. For a more accurate pitch estimate, the frequency portions will have a greater similarity to each other and reversed versions of each other (e.g., each harmonic being centered in a frequency portion) and thus the correlations may be higher. For a less accurate pitch estimate, the frequency portions will have less similarity to each other and reversed versions of each other (e.g., each harmonic being off center by an amount corresponding to the harmonic number) and thus correlations may be lower. 
     Each of the correlations may be computed, for example, by performing an inner product of the two frequency portions (or with a frequency portion and a reversed version of that frequency portion of another frequency portion). The correlation may also be normalized by dividing by N−1 where N is the number of samples in each frequency portion. In some implementations, a Pearson product-moment correlation coefficient may be used. 
     Some or all of the above correlations may be used to determine a score for an accuracy of a pitch estimate. For example, for eight harmonics, eight correlations may be computed for the correlation of a frequency portion with a reversed version of itself, 28 correlations may be computed for a correlation between a frequency portion and another frequency portion, and 28 correlations may be computed between a frequency portion and a reversed version of another frequency portion. These correlations may be combined in any appropriate way to get an overall score for the accuracy of a pitch estimate. For example, the correlations may be added or multiplied to get an overall score. 
     In some implementations, the correlations may be combined using the Fisher transformation. The Fisher transformation of an individual correlation, r, may be computed as 
               F   ⁡     (   r   )       =       1   2     ⁢     log   ⁡     (       1   +   r       1   -   r       )               
In the region of interest for an individual correlation, the Fisher transformation may be approximated as
 
 F ( r )≈ r  
 
The Fisher transformation of an individual correlation may have a probability density function that is approximately Gaussian with a standard deviation of 1/√{square root over (N−3)} where N is the number of samples in each portion. Accordingly, using the above approximation, the probability density function of the Fisher transformation of an individual correlation, f(r), may be represented as
 
               f   ⁡     (   r   )       =           N   -   3       2   ⁢   π         ⁢     e       -     1   2       ⁢     (     N   -   3     )     ⁢     r   2                 
An overall score may then be computed by computing f(r) for each correlation and multiplying them together. Accordingly, if there are M correlations, then an overall score, S, may be computed as a likelihood
 
             S   =       ∏     i   =   1     M     ⁢           ⁢     f   ⁡     (     r   i     )               
or alternatively, the score, S, may be computed as a log likelihood
 
     
       
         
           
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     These scores may be used to obtain a precise pitch estimate through an iterative procedure, such as a golden section search or any kind of gradient descent algorithm. For example, the precise pitch estimate may be initialized with the coarse pitch estimate. A score may be computed for the current precise pitch estimate and for other pitch values near the precise pitch estimate. If the score for another pitch value is higher than the score of the current pitch estimate, then the current pitch estimate may be set to that other pitch value. This process may be repeated until an appropriate stopping condition has been reached. 
     In some implementations, the process of determining the precise pitch estimate may be constrained, for example, by requiring the precise pitch estimate to be within a range of the coarse pitch estimate. The range may be determining using any appropriate techniques. For example, the range may be determined from a variance or a confidence interval of the coarse pitch estimate, such as determining a confidence interval of the coarse pitch estimate using bootstrapping techniques. The range may be determined from the confidence interval, such as a multiple of the confidence interval. In determining the precise pitch estimate, the search may be limited so that the precise pitch estimate never goes outside of the specified range. 
     In some implementations, after determining a fractional chirp rate and a pitch, it may be desired to estimate amplitudes of harmonics of the signal (which may be complex valued and include phase information). Each of the harmonics may be modeled as a chirplet, where the frequency and chirp rate of the chirplet are set using the estimated pitch and estimate fractional chirp rate. For example, for the kth harmonic, the frequency of the harmonic may be k times the estimated pitch, and the chirp rate of the harmonic may be the fractional chirp rate times the frequency of the chirplet. Any appropriate duration may be used for the chirplet. 
     The amplitudes of the harmonics may be estimated using any appropriate techniques, including, for example, maximum likelihood estimation. In some implementations, a vector of harmonic amplitudes, â, may be estimated as
 
{circumflex over ( a )}=( MM   h ) −1   Mx  
 
where M is a matrix where each row corresponds to a chirplet for each harmonic with parameters as described above, the number of rows of the matrix M corresponds to the number of harmonic amplitudes to be estimated, h is a Hermitian transpose, and x is a time series representation of the signal. The estimate of the harmonic amplitudes may be complex valued, and in some implementations, other functions of the amplitudes may be used, such as a magnitude, magnitude squared, real part, or imaginary part.
 
     In some implementations, the amplitudes may have been computed in previous steps and need not be explicitly computed again. For example, where an LLR spectrum is used in previous processing steps, the amplitudes may be computed in computing the LLR spectrum. The LLR spectrum is computed by fitting Gaussians to a spectrum, and one fitting parameter of the Gaussian is the amplitude of the Gaussian. The amplitudes of the Gaussians may be saved during the process of computing the LLR spectrum, and these amplitudes may be recalled instead of being recomputed. In some implementations, the amplitudes determined from the LLR spectrum may be a starting point, and the amplitudes may be refined, for example, by using iterative techniques. 
     The above techniques may be carried out for successive portions of a signal to be processed, such as for a frame of the signal every 10 milliseconds. For each portion of the signal that is processed, a fractional chirp rate, pitch, and harmonic amplitudes may be determined. Some or all of the fractional chirp rate, pitch, and harmonic amplitudes may be referred to as HAM (harmonic amplitude matrix) features and a feature vector may be created that comprises the HAM features. The feature vector of HAM features may be used in addition to or in place of any other features that are used for processing harmonic signals. For example, the HAM features may be used in addition to or in place of mel-frequency cepstral coefficients, perceptual linear prediction features, or neural network features. The HAM features may be applied to any application of harmonic signals, including but not limited to performing speech recognition, word spotting, speaker recognition, speaker verification, noise reduction, or signal reconstruction. 
       FIGS. 11-14  are flowcharts illustrating example implementations of the processes described above. Note that, for the flowcharts described below, the ordering of the steps is exemplary and that other orders are possible, not all steps are required and, in some implementations, some steps may be omitted or other steps may be added. The processes of the flowcharts may be implemented, for example, by one or more computers, such as the computers described below. 
       FIG. 11  is a flowchart showing an example implementation of computing features for a first portion of a signal. At step  1110 , a portion of a signal is obtained. The signal may be any signal for which it may be useful to estimate features, including but not limited to speech signals or music signals. The portion may be any relevant portion of the signal, and the portion may be, for example, a frame of the signal that is extracted on regular intervals, such as every 10 milliseconds. 
     At step  1120 , a fractional chirp rate of the portion of the signal is estimated. The fractional chirp rate may be estimated using any of the techniques described above. For example, a plurality of possible fractional chirp rates may be identified and a score may be computed for each of the possible fractional chirp rates. A score may be computed using a function, such as any of the functions g( ) described above. The estimate of the fractional chirp rate may be determined by selecting a fractional chirp rate corresponding to a highest score. In some implementations, a more precise estimate of fractional chirp rate may be determined using iterative procedures, such as by selecting additional possible fractional chirp rates and iterating with a golden section search or a gradient descent. The function g( ) may take as input any frequency representation of the first portion described above, including but not limited to a spectrum of the first portion, an LLR spectrum of the first portion, a generalized spectrum of the first portion, a frequency chirp distribution of the first portion, or a PVT of the first portion. 
     At step  1130 , a frequency representation of the portion of the signal is computed using the estimated fractional chirp rate. The frequency representation may be any representation of the portion of the signal as a function of frequency. The frequency representation may be, for example, a stationary spectrum, a generalized spectrum, an LLR spectrum, or a row of a PVT. The frequency representation may be computed during the processing of step  1120  and need not be a separate step. For example, the frequency representation may be computed during other processing that determines an estimate of the fractional chirp rate. 
     At step  1140 , a coarse pitch estimate is computed from the portion of the signal using the frequency representation. The coarse pitch estimate may be determined using any of the techniques described above. For example, peak-to-peak distances may be determined for any of the types of spectra described above and for a variety of parameters, such as different thresholds, different smoothing kernels, and from other portions of the signal. The coarse pitch estimate may then be computed from the peak-to-peak distances using a histogram or any of the other techniques described above. 
     At step  1150 , a precise pitch estimate is computed from the portion of the signal using the frequency representation and the coarse pitch estimate. The precise pitch estimate may be initialized with the coarse pitch estimate and then refined with an iterative procedure. For each possible value of a precise pitch estimate, a score, such as a likelihood or a log likelihood, may be computed, and the precise pitch estimate may be determined by maximizing the score. The score may be determined using combinations of correlations as described above. The score may be maximized using any appropriate procedure, such as a golden section search or a gradient descent. 
     At step  1160 , harmonic amplitudes are computed using the estimated fractional chirp rate and the estimated pitch. For example, the harmonic amplitudes may be computed by modeling each harmonic as a chirplet and performing maximum likelihood estimation. 
     The process of  FIG. 11  may be repeated for successive portions or time intervals of the signal. For example, a fractional chirp rate, pitch, and harmonic amplitudes may be computed every 10 milliseconds. The fractional chirp rate, pitch, and harmonic amplitudes may be used for a wide variety of applications, including but not limited to pitch tracking, signal reconstruction, speech recognition, and speaker verification or recognition. 
       FIG. 12  is a flowchart showing an example implementation of computing fractional chirp rate of a portion of a signal. At step  1210 , a portion of a signal is obtained, as described above. 
     At step  1220 , a plurality of frequency representations of the portion of the signal are computed, and the frequency representations may be computed using any of the techniques described above. Each of the frequency representations may correspond to a fractional chirp rate. In some implementations, the frequency representations may be computed (i) from the rows of a PVT, (ii) from radial slices of a frequency-chirp distribution, or (iii) using inner products of the portion of the signal with chirplets where the chirp rate of the chirplet increases with frequency. 
     At step  1230 , a score is computed for each of the frequency representations and each score corresponds to a fractional chirp rate. The score may indicate a match between the fractional chirp rate corresponding to the score and the fractional chirp rate of the portion of the signal. The scores may be computed using any of the techniques described above. In some implementations, the scores may be computed using an auto-correlation of the frequency representations, such as an auto-correlation of the magnitude squared of a frequency representation. The score may be computed from the auto-correlation using any of Fisher information, entropy, Kullback-Leibler divergence, sum of squared (or magnitude squared) values of the auto-correlation, or a sum of squared second derivatives of the auto-correlation. 
     At step  1240 , a fractional chirp rate of the portion of the signal is estimated. In some implementations, the fractional chirp rate is estimated by selecting a fractional chirp rate corresponding to a highest score. In some implementations, the estimate of the fractional chirp rate may be refined using iterative techniques, such as golden section search or gradient descent. The estimated fractional chirp rate may then be used for further processing of the signal as described above, such as speech recognition or speaker recognition. 
       FIG. 13  is a flowchart showing an example implementation of computing a pitch estimate of a portion of a signal. At step  1310 , a first portion of a signal is obtained, as described above, and at step  1320 , a frequency representation of the first portion of the signal is computed, using any of the techniques described above. 
     At step  1330 , a threshold is selected using any of the techniques described above. For example, a threshold may be selected using a signal to noise ratio or may be selected using a height of a peak in the frequency representation of the first portion of the signal. 
     At step  1340 , a plurality of peaks in the frequency representation of the first portion of the signal are identified. The peaks may be identified using any appropriate techniques. For example, the values of the frequency representation may be compared to the threshold to identify a continuous portion of the frequency representation (each a frequency portion) that is always above the threshold. The peak may be identified, for example, by selecting a highest point of the frequency portion, selecting the mid-point between the beginning of the portion and the end of the frequency portion, or fitting a curve (such as a Gaussian) to the frequency portion and selecting the peak using the fit. The frequency representation may accordingly be processed to identify frequency portions that are above the threshold and identify a peak for each frequency portion. 
     At step  1350 , a plurality of peak-to-peak distances in the frequency representation of the first portion of the signal are computed. Each of the peaks may be associated with a frequency value that corresponds to the peak. The peak-to-peak distances may be computed as the difference in frequency values of adjacent peaks. For example, if peaks are present at 230 Hz, 690 Hz, 920 Hz, 1840 Hz (e.g., similar to  931 ,  932 ,  933 , and  934  of  FIG. 9B ), then the peak-to-peak distances may be 460 Hz, 230 Hz, and 920 Hz. 
     Steps  1330 ,  1340 , and  1350  may be repeated for other thresholds, changes to other settings with the same threshold, or changes to other settings with other thresholds. For example, as described above multiple thresholds may be selected using the heights of multiple peaks in the frequency representation, the same threshold or other thresholds may be used with a second frequency representation corresponding to a second portion of the signal (e.g., where the second portion is immediately before or immediately after the first portion), and the same or other thresholds may be used with different smoothing kernels. 
     At step  1360  a histogram of peak-to-peak distances is computed. The histogram may use some or all of the peak-to-peak distances described above. Any appropriate bin width may be used, such as a bin width of 2-5 Hz. 
     At step  1370 , a pitch estimate is determined using the histogram of peak-to-peak distances. In some implementations, the pitch estimate may correspond to the mode of the histogram. In some implementations, multiple histograms may be used to determine the pitch estimate. For example, a plurality of histograms may be computed for a plurality of thresholds (or a plurality of thresholds in combination with other parameters, such as time instances or smoothing kernels), and a preliminary pitch estimate may be determined for each of the plurality of histograms. The final pitch estimate may be determined from the plurality of preliminary pitch estimates, for example, by selecting the most common preliminary pitch estimate. 
       FIG. 14  is a flowchart showing an example implementation of computing a pitch estimate of a portion of a signal. At step  1410 , a frequency representation of a portion of a signal is obtained, as described above. 
     At step  1420 , a pitch estimate of the portion of the signal is obtained. The obtained pitch estimate may have been computed using any technique for estimating pitch, including but not limited to the coarse pitch estimation techniques described above. The obtained pitch estimate may be considered an initial pitch estimate to be updated or may be considered a running pitch estimate that is updated through an iterative procedure. 
     At step  1430 , a plurality of frequency portions of the frequency representation is obtained. Each of the frequency portions may be centered at a multiple of the pitch estimate. For example, a first frequency portion may be centered at the pitch estimate, a second frequency portion may be centered at twice the pitch estimate, and so forth. Any appropriate widths may be used for the frequency portions. For example, the frequency portions may partition the frequency representation, may overlap, or have spaces between them. 
     At step  1440 , a plurality of correlations is computed using the plurality of frequency portions of the frequency representation. The frequency portions may be further processed before computing the correlations. For example, each frequency portion may be extracted from the frequency representation and stored in a vector of length N, where the beginning of the vector corresponds to the beginning of the frequency portion and the end of the vector corresponds to the end of the frequency portion. The frequency portions may be shifted by sub-sample amounts so that the frequency portions line up accurately. For example, the pitch estimate may lie between frequency bins of the frequency representation (e.g., a pitch estimate of 230 Hz may lie between frequency bin  37  and frequency bin  38  with and approximate location of 37.3). Accordingly, the beginning, center, and end of the frequency portions may be defined by fractional sample values. The frequency portions may be shifted by subsample amounts so that one or more of the beginning, center, and end of the frequency portions corresponds to an integer sample of the frequency representation. In some implementations, the frequency portions may also be normalized by subtracting a mean and dividing by a standard deviation of the frequency portion. 
     The correlations may include any of a correlation between a first frequency portion and a second frequency portion, a correlation between a first frequency portion and a reversed second frequency portion, and a correlation between a first frequency portion and a reversed first frequency portion. The correlations may be computed using any appropriate techniques. For example, the frequency portions may be extracted from the frequency representation and stored in a vector, as described above, and the correlations may be computed by performing inner products of the vectors (or an inner product of a vector with a reversed version of another vector). 
     At step  1450 , the correlations are combined to obtain a score for the pitch estimate. Any appropriate techniques may be used to generate a score, including for example, computing a product of the correlations, a sum of the correlations, a combination of the Fisher transformation of the correlations, or a combination likelihoods or log-likelihoods of the correlations or Fisher transformation of the correlations, as described above. 
     At step  1460 , the pitch estimate is updated. For example, a first score for a first pitch estimate may be compared to a second score for a second pitch estimate, and the pitch estimate may be determined by selecting the pitch estimate with a highest score. Steps  1420  to  1460  may be repeated to continuously update a pitch estimate using techniques such golden section search or gradient descent. Steps  1420  to  1460  may be repeated until some appropriate stop condition has been reached such as a maximum number of iterations or the improvement in the pitch estimate from a previous estimate falling below a threshold. 
       FIG. 15  illustrates components of one implementation of a computing device  110  for implementing any of the techniques described above. In  FIG. 15 , the components are shown as being on a single computing device  1510 , but the components may be distributed among multiple computing devices, such as a system of computing devices, including, for example, an end-user computing device (e.g., a smart phone or a tablet) and/or a server computing device (e.g., cloud computing). For example, the collection of audio data and pre-processing of the audio data may be performed by an end-user computing device and other operations may be performed by a server. 
     Computing device  1510  may include any components typical of a computing device, such as volatile or nonvolatile memory  1520 , one or more processors  1521 , and one or more network interfaces  1522 . Computing device  1510  may also include any input and output components, such as displays, keyboards, and touch screens. Computing device  1510  may also include a variety of components or modules providing specific functionality, and these components or modules may be implemented in software, hardware, or a combination thereof. Below, several examples of components are described for one example implementation, and other implementations may include additional components or exclude some of the components described below. 
     Computing device  1510  may have a signal processing component  1530  for performing any needed operations on an input signal, such as analog-to-digital conversion, encoding, decoding, subsampling, windowing, or computing frequency representations. Computing device  1510  may have a fractional chirp rate estimation component  1531  that estimates fractional chirp rate of a signal using any of the techniques described above. Computing device  1510  may have a coarse pitch estimation component  1532  that estimates the pitch of a signal using peak-to-peak distances as described above. Computing device  1510  may have a precise pitch estimation component  1533  that estimates the pitch of a signal using correlations as described above. Computing device  1510  may have a HAM feature generation component  1534  that determines amplitudes of harmonics as described above. 
     Computing device  1510  may also have components for applying the above techniques to particular applications. For example, computing device  1510  may have any of a speech recognition component  1540 , a speaker verification component  1541 , a speaker recognition component  1542 , a signal reconstruction component  1543 , and a word spotting component  1544 . For example, any of an estimated fractional chirp rate, an estimated pitch, and estimated harmonic amplitudes may be used as input to any of the applications and used in addition to or in place of other features or parameters used for these applications. 
     Depending on the implementation, steps of any of the techniques described above may be performed in a different sequence, may be combined, may be split into multiple steps, or may not be performed at all. The steps may be performed by a general purpose computer, may be performed by a computer specialized for a particular application, may be performed by a single computer or processor, may be performed by multiple computers or processers, may be performed sequentially, or may be performed simultaneously. 
     The techniques described above may be implemented in hardware, in software, or a combination of hardware and software. The choice of implementing any portion of the above techniques in hardware or software may depend on the requirements of a particular implementation. A software module or program code may reside in volatile memory, non-volatile memory, RAM, flash memory, ROM, EPROM, or any other form of a non-transitory computer-readable storage medium. 
     Conditional language used herein, such as, “can,” “could,” “might,” “may,” “e.g.,” is intended to convey that certain implementations include, while other implementations do not include, certain features, elements and/or steps. Thus, such conditional language indicates that that features, elements and/or steps are not required for some implementations. The terms “comprising,” “including,” “having,” and the like are synonymous, used in an open-ended fashion, and do not exclude additional elements, features, acts, operations. The term “or” is used in its inclusive sense (and not in its exclusive sense) so that when used, for example, to connect a list of elements, the term or means one, some, or all of the elements in the list. 
     Conjunctive language such as the phrase “at least one of X, Y and Z,” unless specifically stated otherwise, is to be understood to convey that an item, term, etc. may be either X, Y or Z, or a combination thereof. Thus, such conjunctive language is not intended to imply that certain embodiments require at least one of X, at least one of Y and at least one of Z to each be present. 
     While the above detailed description has shown, described and pointed out novel features as applied to various implementations, it can be understood that various omissions, substitutions and changes in the form and details of the devices or techniques illustrated may be made without departing from the spirit of the disclosure. The scope of inventions disclosed herein is indicated by the appended claims rather than by the foregoing description. All changes which come within the meaning and range of equivalency of the claims are to be embraced within their scope.