white noise plus echo

Here are two relevant references:

Bilsen, F.A. (1966). Repetition pitch: Monaural interaction of a sound with
the repetition of the same, but phase shifted, sound. Acustica 17, 295-300

When a sound and the repetition of the same sound after a delay $\tau$ are
presented together, monaurally, one perceives a "coloration", accompanied by
a pitch sensation. This pitch, RP (Repetition Pitch), appears to correspond
to the reciprocal $1/\tau$ of the delay time $\tau$. However, when all the
frequency components of the delayed sound are shifted in phase by
$90^\circ$, $180^\circ$ and $270^\circ$ resp., the RP changes to
$1.07/\tau$, $0.88/\tau$ (or $1.14/\tau$) and $0.94/\tau$ resp. (In our case
the sound was white noise, delayed by a tape recorder.) The same pitches are
perceived when listening to a random sequence of pulse pairs (pulse distance
$\tau$), where the first of (and) the second pulse is shifted in phase.
These results cannot be explained simply by means of autocorrelation or
frequency analysis.

Yost, W.A., Hill, R., Perez-Falcon, T. (1978). Pitch and pitch
discrimination of broadband signals with rippled power spectra. J. Acoust.
Soc. Am. 64, 1166-1173

A random-interval pulse train or wide-band noise when delayed ($\tau$) and
added back to itself (cos+) produces a stimulus with a cosinusoidally
varying (or rippled) power spectrum. The spacing between the peaks in the
spectrum is equal to the reciprocal of the delay ($1/\tau$). If the stimulus
is delayed and added back at $180^\circ$ phase reversal (cos-), then a
cosinusoidally varying power spectrum is generated whose spectral peaks are
separated by $1/\tau$, but whose peaks are displaced by $1/2\tau$ relative
to the power spectrum of the cos+ stimulus generated with the same delay,
$\tau$. These stimuli yield a pich, such that the pitch of the cos+ stimulus
is equal to approximately $1/\tau$ and the pitches of the cos- stimuli are
equal to approximately $0.9/\tau$ and $1.1/\tau$. These pitch matching
results were studied using a variety of matching stimuli and conditions.
Following the identification of the pitches, a method of limits and a
same-different procedure were used to study the pitch discriminability of
both the cos+ and cos- stimuli. Delays ($\tau$) ranging from 1 to 10 ms were
studied covering a pitch range of 90-1100 Hz. The pitch discrimination
associated with the cos+ and cos- stimuli were essentially the same for both
the random-interval pulse train and the wide-band stimuli. These
pitch-discrimination results are compared to those associated with a
periodic pulse train. The research is also discussed in terms of
discriminations of delayed sounds in reverberant environments. These results
are consistent with assumptions concerning the autocorrelation of the
rippled stimuli within the dominant frequency region for pitch perception.