Heterodyning is based on the trigonometric identity:
\sin \theta \sin \varphi = \frac{1}{2}\cos(\theta - \varphi) - \frac{1}{2}\cos(\theta + \varphi)
The product on the left hand side represents the multiplication ("mixing") of a sine wave with another sine wave. The right hand side shows that the resulting signal is the difference of two sinusoidal terms, one at the sum of the two original frequencies, and one at the difference, which can be considered to be separate signals.
Using this trigonometric identity, the result of multiplying two sine wave signals, \sin (2 \pi f_1 t)\, and \sin (2 \pi f_2 t)\, can be calculated:
\sin (2 \pi f_1 t)\sin (2 \pi f_2 t) = \frac{1}{2}\cos 2 \pi (f_1 - f_2) t - \frac{1}{2}\cos 2 \pi (f_1 + f_2) t \,
The result is the sum of two sinusoidal signals, one at the sum f1 + f2 and one at the difference f1 - f2 of the original frequencies
The two signals are multiplied in the mixer. In order to multiply the signals, the mixer must be a nonlinear component, that is, its output current or voltage must be a nonlinear function of its input. Most circuit elements in communications circuits are designed to be linear. This means they obey the superposition principle; if F(v) is the output of a linear element with an input of v:
F(v_1 + v_2) = F(v_1) + F(v_2) \,
So if two sine wave signals are applied to a linear device, the output is simply the sum of the outputs when the two signals are applied separately, with no product terms. So the function F must be nonlinear. Examples of nonlinear components that are used as mixers are vacuum tubes and transistors biased near cutoff (class C), and diodes. For lower frequencies, IC analog multipliers can be used which multiply signals precisely. Ferromagnetic core inductors driven into saturation can also be used. In nonlinear optics, crystals that have nonlinear characteristics are used to mix laser light beams to create heterodynes at optical frequencies.
To demonstrate mathematically how a nonlinear component can multiply signals and generate heterodyne frequencies, the nonlinear function F can be expanded in a power series (MacLaurin series):
F(v) = \alpha_1 v + \alpha_2 v^2 + \alpha_3 v^3 + \ldots \,
To simplify the math, the higher order terms above α2 will be indicated by an ellipsis (". . .") and only the first terms will be shown. Applying the two sine waves at frequencies ω1 = 2πf1 and ω2 = 2πf2 to this device:
v_{out} = F(A_1 \sin \omega_1 t + A_2 \sin \omega_2 t)\,
v_{out} = \alpha_1 (A_1 \sin \omega_1 t + A_2 \sin \omega_2 t) + \alpha_2(A_1 \sin \omega_1 t + A_2 \sin \omega_2 t)^2 + \ldots \,
v_{out} = \alpha_1 (A_1 \sin \omega_1 t + A_2 \sin \omega_2 t) + \alpha_2(A_1^2 \sin^2 \omega_1 t + 2 A_1 A_2 \sin \omega_1 t \sin \omega_2 t + A_2^2 \sin^2 \omega_2 t) + \ldots \,
It can be seen that the second term above contains a product of the two sine waves. Simplifying with trigonometric identities:
v_{out} = \alpha_1 (A_1 \sin \omega_1 t + A_2 \sin \omega_2 t) + \alpha_2(\frac{A_1^2}{2} 1 - \cos 2 \omega_1 t + A_1 A_2 \cos (\omega_1 t - \omega_2 t) - \cos (\omega_1 t + \omega_2 t) + \frac{A_2^2}{2} 1 - \cos 2 \omega_2 t ) + \ldots \,
v_{out} = \alpha_2 A_1 A_2 \cos (\omega_1 - \omega_2 )t - \alpha_2 A_1 A_2 \cos (\omega_1 + \omega_2 ) t + \ldots \,
So the output contains sinusoidal terms with frequencies at the sum ω1 + ω2 and difference ω1 - ω2 of the two original frequencies. It also contains terms at the original frequencies and at multiples of the original frequencies 2ω1, 2ω2, 3ω1, 3ω2, etc.; the latter are called harmonics. These unwanted frequencies, al
\sin \theta \sin \varphi = \frac{1}{2}\cos(\theta - \varphi) - \frac{1}{2}\cos(\theta + \varphi)
The product on the left hand side represents the multiplication ("mixing") of a sine wave with another sine wave. The right hand side shows that the resulting signal is the difference of two sinusoidal terms, one at the sum of the two original frequencies, and one at the difference, which can be considered to be separate signals.
Using this trigonometric identity, the result of multiplying two sine wave signals, \sin (2 \pi f_1 t)\, and \sin (2 \pi f_2 t)\, can be calculated:
\sin (2 \pi f_1 t)\sin (2 \pi f_2 t) = \frac{1}{2}\cos 2 \pi (f_1 - f_2) t - \frac{1}{2}\cos 2 \pi (f_1 + f_2) t \,
The result is the sum of two sinusoidal signals, one at the sum f1 + f2 and one at the difference f1 - f2 of the original frequencies
The two signals are multiplied in the mixer. In order to multiply the signals, the mixer must be a nonlinear component, that is, its output current or voltage must be a nonlinear function of its input. Most circuit elements in communications circuits are designed to be linear. This means they obey the superposition principle; if F(v) is the output of a linear element with an input of v:
F(v_1 + v_2) = F(v_1) + F(v_2) \,
So if two sine wave signals are applied to a linear device, the output is simply the sum of the outputs when the two signals are applied separately, with no product terms. So the function F must be nonlinear. Examples of nonlinear components that are used as mixers are vacuum tubes and transistors biased near cutoff (class C), and diodes. For lower frequencies, IC analog multipliers can be used which multiply signals precisely. Ferromagnetic core inductors driven into saturation can also be used. In nonlinear optics, crystals that have nonlinear characteristics are used to mix laser light beams to create heterodynes at optical frequencies.
To demonstrate mathematically how a nonlinear component can multiply signals and generate heterodyne frequencies, the nonlinear function F can be expanded in a power series (MacLaurin series):
F(v) = \alpha_1 v + \alpha_2 v^2 + \alpha_3 v^3 + \ldots \,
To simplify the math, the higher order terms above α2 will be indicated by an ellipsis (". . .") and only the first terms will be shown. Applying the two sine waves at frequencies ω1 = 2πf1 and ω2 = 2πf2 to this device:
v_{out} = F(A_1 \sin \omega_1 t + A_2 \sin \omega_2 t)\,
v_{out} = \alpha_1 (A_1 \sin \omega_1 t + A_2 \sin \omega_2 t) + \alpha_2(A_1 \sin \omega_1 t + A_2 \sin \omega_2 t)^2 + \ldots \,
v_{out} = \alpha_1 (A_1 \sin \omega_1 t + A_2 \sin \omega_2 t) + \alpha_2(A_1^2 \sin^2 \omega_1 t + 2 A_1 A_2 \sin \omega_1 t \sin \omega_2 t + A_2^2 \sin^2 \omega_2 t) + \ldots \,
It can be seen that the second term above contains a product of the two sine waves. Simplifying with trigonometric identities:
v_{out} = \alpha_1 (A_1 \sin \omega_1 t + A_2 \sin \omega_2 t) + \alpha_2(\frac{A_1^2}{2} 1 - \cos 2 \omega_1 t + A_1 A_2 \cos (\omega_1 t - \omega_2 t) - \cos (\omega_1 t + \omega_2 t) + \frac{A_2^2}{2} 1 - \cos 2 \omega_2 t ) + \ldots \,
v_{out} = \alpha_2 A_1 A_2 \cos (\omega_1 - \omega_2 )t - \alpha_2 A_1 A_2 \cos (\omega_1 + \omega_2 ) t + \ldots \,
So the output contains sinusoidal terms with frequencies at the sum ω1 + ω2 and difference ω1 - ω2 of the two original frequencies. It also contains terms at the original frequencies and at multiples of the original frequencies 2ω1, 2ω2, 3ω1, 3ω2, etc.; the latter are called harmonics. These unwanted frequencies, al
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