What is a "pitch period"?

Question

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!

Answer 1

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.

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:

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

Answer 2

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 $60$.