Complex
Sounds
Reading: Yost Ch. 4
Natural Sounds
Most sounds in our everyday lives are not
simple sinusoidal sounds, but are complex
sounds, consisting of a sum of many
sinusoids.
The amplitude and frequency content of
natural sounds varies with time.
A spectrogram is a visual representation
of the spectrum of frequencies in a sound
(or other signal) as they vary with time.
Time vs. Frequency Domain
There are two ways to describe a sound
wave: in the time domain and in the
frequency domain.
• Time domain: describes how the
instantaneous sound pressure/intensity
varies as a function of time.
• Frequency domain: describes sounds in
terms of the individual sinusoids that are
added together to produce the sound, in
the form of amplitude- and phase
spectra as a function of frequency.
Fourier Analysis 1
Pr
...
Fourier analysis can be used to convert
between the time and frequency domain
representations of a sound. In particular,
Fourier’s theorem states that all time domain
sounds are composed of a sum of sinusoids
of different frequencies, amplitudes and
phases.
•
•
If the signal is periodic, then the frequencies of
the sinusoids are harmonically related (integer
multiples of the fundamental frequency, f0=1/Pr)
If the signal is aperiodic, then the frequencies
are continuous
line spectrum
continuous
spectrum
f0 = 1/Pr
frequency (Hz)
Fourier Analysis 2
If the amplitude variation of a complex sound is
important, then the waveform is described in
the time domain. If the frequency content is
important, then the amplitude and phase
spectra are described in the frequency domain.
One representation is obtained from the other
(time-frequency) using the Fourier transform
or inverse Fourier transform , respectively.
The first term in the Fourier
series for the square wave is
shown in purple.
The second term in the Fourier
series for the square wave is
shown in purple; the sum is red.
The third term in the Fourier
The sum of the first 10 terms is
series for the square wave is
shown in red. It looks very like a
shown in purple; the sum is red. square wave with some bumpiness.
Tone Bursts
• Tone burst: “gated” sinusoid with discrete
onsets and offsets.
• Spectrum of TB is continuous: extends over
a wider frequency range than infinitely long
tone (which has a line spectrum).
• Onsets and offsets add components, cause
“spectral splatter”
• Splatter around tone frequency heard as
onset and offset clicks when tone burst is
brief.
• Tone burst more clearly tonal and less clicklike the longer it is on, because the
proportion of Etotal at the frequency of
dominant sinusoid increases with duration.
• Splatter reduced by “shaping” rise/fall time
with gradual onsets and offsets to make it
inaudible.
Amplitude and Frequency Modulation
Two direct ways to change a simple
stimulus into a complex one are to
alter the amplitude and frequency of
the sinusoid with time.
Modulation means varying some
aspect of a continuous wave carrier
with an information-bearing
modulation waveform.
• In amplitude modulation (AM), the
amplitude or "strength" of the
carrier oscillations is varied with
time.
• In frequency modulation (FM), the
instantaneous frequency of the
carrier is varied with time.
Sinusoidal Amplitude Modulation 1
Amplitude of a signal changes in a
sinusoidal manner over time (SAM).
Start with a sine-wave “carrier” (where
fc = carrier frequency)
10
0
D(t) = A sin(2πfct)
Let carrier amplitude (A) vary in a
sinusoidal manner over time:
A(t) = [1 + m sin(2πfmt)]
where fm = modulation frequency, and
m = modulation amplitude (0 ≤ m ≤ 1).
Therefore: D(t) = (A •A(t)) sin(2πfct)
D(t) = A [1 + m sin(2πfmt)] sin(2πfct)
2
1
0
m
Sinusoidal Amplitude Modulation 2
Top: SAM tone, showing “fine structure”
(waveform) of carrier (fc) varying in
amplitude at a modulation rate of fm.
Bottom: Amplitude (left) and phase (right)
spectra of SAM tone above.
Amplitude spectrum: fc is flanked by
two sideband frequencies at
(fc + fm) and (fc – fm).
Sideband amplitudes are equal to
A(m/2), where m is the amplitude of
modulator.
tone
SAM
tone
90
0
-90
Frequency Modulation
Frequency (f) of a signal varies with time; often
in a linear (LFM) or sinusoidal manner (SFM).
LFM
LFM: f varies in a linear manner with time from
a starting frequency to an ending frequency in
an interval t:
SFM
SFM
D(t) = A sin(2πft)
f = fstart + kt, where k = (fend – fstart) / T
The parameter k is call the chirp rate.
SFM: f varies in a sinusoidal manner in time
f = fc + m sin(2πfmt), where m = fmax –fmin
M is the modulation depth and fm is the
modulation rate (δf/t in Hz/s)
The spectra of FM stimuli changes over time;
changes in frequency (and amplitude) over time
can be graphed using spectrograms.
“Spectrogram” AM/FM
Beats
The addition of two sinusoids of different
frequencies also produces a complex
sound. If the two tones have very different
frequencies, then they add like harmonics
in a series (where the frequencies can be
seen as the time separations (1/f1 and
1/f2) between peaks).
If the tones are close in frequency, then
the waveform appears to be a single tone
with a sinusoidal amplitude modulation
(similar to, but not the same, as SAM)
• Frequency = mean = (f1+f2)/2
• Amplitude of waveform beats at a
rate equal to f1-f2
1/f1
1/f2
1/(f1-f2)
1/[(f1+f2)]/2
Square Waves
A square wave is equivalent to turning a
sound on and off in a periodic manner
such that the on time is the same as the
off time. Pulse on time = pulse off time
(50% “duty cycle”).
Pr = 2 ms
Square waves have spectral energy at a
fundamental frequency f0 = 1/Pr, and at
odd harmonics of f0: 3f0, 5f0, 7f0 etc..
Amplitude of each higher harmonic =
1/harmonic number:
3rd harmonic (3f0) = 1/3(Af0)
5th harmonic (5f0) = 1/5(Af0), etc.
F0 = 500 Hz
1500 Hz
2500 Hz
Transients
In studying hearing, it is often useful to
present a very brief acoustic click. Transients
(clicks) are very brief, non-sinusoidal
“impulses” that are the sum of many
sinusoids in phase at only one point in time.
Click spectra are distinctive:
• Amplitude spectrum: Continuous
with minima at frequencies equal
to integer multiples of 1/D, where
D = click duration in sec.
• Phase spectrum: 90° for all
frequencies.
D
1/D
f = 1/D
2/D
3/D
Click Trains
Click trains
• clicks repeating at a constant rate with
period Pr
Generate a line spectrum (like a harmonic
series) with discrete frequencies equal to
integer multiples of 1/Pr (i.e., the inter-click
interval).
Click duration (D) shapes the spectrum,
generating spectral notches at frequency of
1/D, 2/D, 3/D, etc.
“Repetition Pitch”: Click trains can give rise
to pitch perception because the temporal
structure and spectrum are similar to complex
tones.
scale
Repetition Pitch
Repetition pitch is the sensation of tonality
in a sound, in which this tonal quality is
solely obtained due to a repeated pattern,
rather than a sinusoidal waveform.
The effect is perceived most prominently if
the repeated sound contains a wide
spectrum of frequencies, like clicks.
“Voiced” sounds (e.g., vowels) are
basically click trains produced by larynx,
with spectra shaped by vocal tract.
Amplitude spectra of three
vowels in human speech
Noise
By definition, noise is a sound whose
instantaneous amplitude (A) varies
randomly in time.
Gaussian noise: A varies probabilistically
according to normal (“Gaussian”)
distribution.
Phase also varies randomly in Gaussian
noise.
Noise Intensity
•
Total noise power (TP): Sum of
amplitudes of all sinusoids (i.e.,
bandwidth x intensity).
•
“Spectrum level” (N0):
Average power per unit bandwidth
(i.e., average intensity in a band of
noise 1 Hz wide):
– N0 = TP/BW
= TP (dB) – 10 log BW
– For the figure:
N0 (dB) = 50 dB
= 80 dB – 10 log(1000)
“Colors” of Noise
•
White noise:
– power spectrum is flat across all
frequencies.
– Bandwidth is very broad.
•
Pink noise:
– Average power level drops 50%
(3 dB) per octave.
– Therefore, power level is constant
ratio of center frequency to BW.
Narrowband Noise
Noises are usually broadband, containing
a large range of frequencies, but can also
be narrowband containing a limited
number of frequency components.
Narrowband noises have an envelope that
fluctuates in proportion to noise bandwidth.
That is, the rate of amplitude modulation
increases with increasing bandwidth:
• Left (a): BW = 10 Hz
• Right (b): BW = 25 Hz
Narrow-band
(band limited) noise
Envelopes
Many signals can be characterized by
an envelope and the fine-structure
waveform that falls under the envelope
(e.g., SAM tones). In fact, most
waveforms can be described by the
following formula:
x(t) = e(t)f(t)
where e(t) is the envelope function, f(t)
is the fine-structure waveform, and x(t)
is the complex waveform.
SAM noise:
x(t) = [1 + m sin(2πFmt)] n(t)