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The term pitch period appears in the book Speech and Language Processing by Daniel Jurafsy:

As we just said, a pitch-synchronous algorithm is one in which we do something at each pitch period or epoch. For such algorithms it is important to have very accurate pitch markings: measurements of exactly where each pitch pulse or epoch occurs. An epoch can be defined by the instant of maximum glottal pressure, or alternatively by the instant of glottal closure.

I have some difficulties to understand what a pitch period means here. Could someone please explain more about the concept of pitch period to me? Visualize an example of a pitch period in a graph like waveform or spectragram would be more helpful.

Thanks in advance!

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2 Answers 2

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So based in @rbj answer, can i try to show you visually ...

Take a look in this pure tone signal, one sine at 100hz and sample rate at 44100hz.

enter image description here

just by looking you can find where the pattern starts to repeat itself, I marked it with the naked eye as being in position 441(Period)

So we take exactly 100hz --> 44100/441=100hz

Another example:

enter image description here

Another 100hz signal, it was sampled at 48000hz, one more time just looking the wavform signal i try mark where this start to be periodic, for my eyes i marked it at position 447(Period)

So it give me 48000/477=±100hz to be exact = 100.6289hz

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  • $\begingroup$ thanks!Is the term pitch period just the same as the term period ? $\endgroup$ Commented Jul 13, 2021 at 14:50
  • $\begingroup$ pitch and period can be equivalent terms, if you have the period you will be able to turn it into pitch and vice versa $\endgroup$
    – ederwander
    Commented Jul 13, 2021 at 16:56
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The pitch period of a perfectly periodic function, $x(t)$, is the smallest positive value $P>0$ such that

$$ x(t+P) = x(t) \qquad \forall t \in \mathbb{R} $$

Now, simply because a function is periodic with period $P$, then it is also periodic with periods $2P$ or $3P$ or $4P$ or any integer multiple of $P$, but we don't pick $2P$ or $3P$ or $4P$ for the period of $x(t)$, we pick the smallest possible $P$ that is a positive value.

The reciprocal of that period $P$ is the fundamental frequency

$$ f_0 \triangleq \frac{1}{P} $$

From that, it is possible to represent the period signal $x(t)$ as a Fourier Series

$$\begin{align} x(t) &= \sum\limits_{k=-\infty}^{\infty} c_k \ e^{j 2 \pi k f_0 t} \\ \\ &= c_0 + \sum\limits_{k=1}^{\infty} |c_k| \cos(2 \pi k f_0 t + \phi_k) \qquad \phi_k = \arg\{c_k\} \\ \\ &= c_0 + \sum\limits_{k=1}^{\infty} \big(\underbrace{|c_k|\cos(\phi_k)}_{a_k}\big) \cos(2 \pi k f_0 t) - \big(\underbrace{|c_k|\sin(\phi_k)}_{b_k}\big) \sin(2 \pi k f_0 t) \\ \\ &= c_0 + \sum\limits_{k=1}^{\infty} a_k \cos(2 \pi k f_0 t) - b_k \sin(2 \pi k f_0 t) \\ \end{align}$$

where $$ c_k = a_k + j\,b_k = \int\limits_{t_0}^{t_0+P} x(t) e^{-j 2 \pi k f_0 t} \ \mathrm{d}t \qquad t_0 \in \mathbb{R} $$

Now most audio that sounds like a note or a tone, including voiced speech, is not exactly periodic, but is "quasi-periodic". So our defining condition of a quasiperiodic $x(t)$ is

$$ x\big(t+P(t)\big) \approx x(t) \qquad \forall t \in \mathbb{R} $$

Now, instead of constant $P$, the period is a virtually constant (or slowly varying) function of time $P(t)$. The fundamental frequency then would also be a virtually constant function of time

$$ f_0(t) \triangleq \frac{1}{P(t)} $$

Now the pitch of that tone or note is the logarithm of the fundamental frequency $f_0$. If that pitch is expressed in units of octaves, it's the base-2 logarithm. If the pitch is expressed in units of semitones, it's 12 times that base-2 logarithm. The pitch has to be relative to a standard pitch. For MIDI, the pitch is

$$ p(t) = 12 \log_2\left( \frac{f_0(t)}{440 \, \mathrm{Hz}} \right) + 69 $$

and is in units of semitones with Middle C having a pitch of $60$.

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  • $\begingroup$ thanks!Is the term pitch period just the same as the term period $\endgroup$ Commented Jul 13, 2021 at 14:49
  • $\begingroup$ As far as i can tell. It's also might not be a constant value like you would see in a perfectly periodic tone. "period" might imply a constant value. "pitch period" might imply a slowly varying value just as pitch is slowly varying. $\endgroup$ Commented Jul 13, 2021 at 16:54

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