This is an amplitude modulated waveform, in which a low frequency signal with beat frequency ωb modulates a high frequency signal with carrier frequency ω0 rad/s. Over short periods of time, the modulating signal cos(ωb t + ψ) remains relatively constant while the carrier term cos(ω0 + φ) produces many cycles of its pure tone. Thus, we perceive the pure tone at the carrier frequency ω0 , having an amplitude that varies sinusoidally at the beat frequency ωb . The following is the script to simulate beating tones in which ω0 = 2π×1400 rad/s and ωb = 2π×100 rad/s, resulting. | This is an amplitude modulated waveform in which a low frequency signal with beat frequency Wb modulates a high frequency signal with carrier frequency Wo rad s. Over short periods of time the modulating signal cos wbt remains relatively constant while the carrier term cos w0 Ộ produces many cycles of its pure tone. Thus we perceive the pure tone at the carrier frequency w0 having an amplitude that varies sinusoidally at the beat frequency Wb. The following is the script to simulate beating tones in which w0 2n X1400 rad s and Wb 2n X100 rad s resulting in the plot shown in Figure . beating sinusoidal tones t linspace -1e-2 1e-2 1001 x cos 2 pi 1500 t cos 2 pi 1300 t m 2 cos 2 pi 100 t plot t m b t -m b t x k . axis . title Beating between tones . xlabel Time s . ylabel Amplitude Figure Beating between tones Polynomials A polynomial is a function of a single variable that can be expressed in the general form A s aS a2sN-1 a3sN 2 - aN s aN 1 111 where the variable is s and the polynomial coefficients are the N 1 constants a1 a2 . aN 1. The polynomial is of degree N the largest value used as an exponent. The general form of a degree 3 cubic polynomial is A s a1s3 a2s2 a3s a4 and a specific example of a cubic polynomial is A s s3 4s3 - 7s - 10 Note that the notation used here is nonstandard as the coefficient of term sk is usually denoted as ak. However the nonstandard notation is more compatible with the indexing of arrays in Matlab as will be explained below. For information on Matlab functions supporting polynomial computations type help polyfun. Polynomial Evaluation There are several ways to evaluate a polynomial for a set of values. Consider the cubic polynomial A s s3 4s2 - 7s - 10 Scalar s use scalar operations A s 3 4ss 2 - 7ss - 10 Vector or matrix s use array or element-by-element operations A s. 3 4ss. 2 - 4ss - 10 The size of the vector or matrix A will be the same as that of s. Using polyval a s Evaluates a polynomial with
