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, f 0 =1/Pr) If the signal is aperiodic, then the frequencies are continuous line spectrum continuous spectrum f 0 = 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 series for the square wave is shown in purple; the sum is red. The sum of the first 10 terms is shown in red. It looks very like a 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 E total 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 f c = carrier frequency) D (t) = A sin(2πf c t) Let carrier amplitude (A) vary in a sinusoidal manner over time: A (t) = [1 + m sin(2πf m t)] where f m = modulation frequency, and m = modulation amplitude (0 m 1). Therefore: D (t) = (A A (t) ) sin(2πf c t) D (t) = A [1 + m sin(2πf m t)] sin(2πf c t) 10 0 2 1 0 m
Sinusoidal Amplitude Modulation 2 Top: SAM tone, showing fine structure (waveform) of carrier (f c ) varying in amplitude at a modulation rate of f m. tone SAM tone 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. 90 0-90
Frequency Modulation Frequency (f) of a signal varies with time; often in a linear (LFM) or sinusoidal manner (SFM). LFM: f varies in a linear manner with time from a starting frequency to an ending frequency in an interval t: D(t) = A sin(2πft) f = f start + kt, where k = (f end f start ) / T The parameter k is call the chirp rate. SFM: f varies in a sinusoidal manner in time LFM SFM Spectrogram AM/FM f = f c + m sin(2πf m t), where m = f max f min M is the modulation depth and f m 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.
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 = (f 1 +f 2 )/2 Amplitude of waveform beats at a rate equal to f 1 -f 2 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 f 0 = 1/Pr, and at odd harmonics of f 0 : 3f 0, 5f 0, 7f 0 etc.. Amplitude of each higher harmonic = 1/harmonic number: 3 rd harmonic (3f 0 ) = 1/3(A f0 ) 5 th harmonic (5f 0 ) = 1/5(A f0 ), etc. F 0 = 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). scale 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.
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 (N 0 ): Average power per unit bandwidth (i.e., average intensity in a band of noise 1 Hz wide): N 0 = TP/BW = TP (db) 10 log BW For the figure: N 0 (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(2πf m t)] n(t)