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GNSS Signals and Range Determination
Published in Basudeb Bhatta, Global Navigation Satellite Systems, 2021
The carrier phase observable is sometimes called the reconstructed carrier phase or carrier beat phase observable. In this context, a beat is the pulsation resulting from the combination of two waves with different frequencies. It can occur when any pair of oscillations with different frequencies are combined. In GNSS, a beat is created when a carrier generated in a receiver and the carrier received from the satellite are combined. At first, that might not seem sensible. How could a beat be created by combining two absolutely identical carriers? It is known to us that there should not be any difference in frequency between a carrier generated in a satellite and a replica carrier generated in a receiver. If there is no difference in the frequencies, how can there be a beat? But there is a slight difference between these two carriers. Something happens to the frequency of the carrier on its trip from a GNSS satellite to a receiver and its frequency changes. The phenomenon is described as the Doppler effect.
Superposition of Waves
Published in Myeongkyu Lee, Optics for Materials Scientists, 2019
The low-frequency wave Eo(x, t) in eq 3.22 serves as an envelope modulating the amplitude of the high-frequency (ωa) wave, as shown in Figure 3.8. The dashed line in Figure 3.8c depicts the envelope of the resultant wave. The energy flux density carried by a wave is proportional to the square of its amplitude. Therefore, the combination of two waves of slightly different frequency exhibits beats.Eo2(x,t) oscillates with a frequency of 2ωs=ω1−ω2. This is known as the beat frequency. The beat frequency is twice the frequency of the modulating envelope. We find that the beat frequency is simply the difference between the frequencies of the two constituent waves.
Background
Published in Russ Martin, Sound Synthesis and Sampling, 2012
If the difference in frequency between the two waveforms is increased, then the speed of the beats will increase. When the frequency of the beats is above 20 Hz, then the mixed sound begins to sound like two separate frequencies. As the difference increases, the two frequencies will pass through a series of ratios of frequency, some of them sounding pleasant to the ear, and others sounding unpleasant. The ratio between the two frequencies is called an interval; the easiest and most ‘pleasing’ interval is a ratio of 2:1, an octave.
Musical sonification supports visual discrimination of color intensity
Published in Behaviour & Information Technology, 2019
The third sonification condition used dissonance of the harmonic content of each tone in the sonification (hereafter referred to as Harmony). A more complex harmonic sound is more captivating for a listener compared to a simpler harmonic sound (Iakovides et al. 2004), and dissonant chords are experienced as more unpleasant compared to harmonious major or minor chords (Pallesen et al. 2005). In this sonification condition, the triangle waves creating each tone varied from seven tones in unison (perfect pitch) in the area with the highest intensity level to almost a halftone below and above the fundamental tone (, , , 0, , , cents) in the lowest intensity area. For the Harmony condition the mapping was done linearly. As the harmonic components are further apart in frequency in relation to the fundamental frequency (as is the case in the darker areas in the visual representation) the interference between frequencies creates a beating (Winckel 1967). The beat frequency is equal to the difference in frequency of the notes that interfere (Roberts 2016). Perception of the two tones ranges from pleasant beating (when there is a small frequency difference) to roughness (when the difference grows larger) and eventually separation into two tones (when the frequency difference increases even more) (Sethares 2005). As a consequence, the beating decreases in tempo as the harmonic components come closer in frequency, and at the brightest vertical pixel column the beating stops and all harmonic components lock to the fundamental frequency. The physical behaviour of the frequencies involved creates a clear sonification clue that makes even small differences in harmonic content rather easily detectable. Consequently, the harmonic complexity of the sonification should provide sonic cues for the participants to solve the tasks in the test.