Summing two frequencies: When two signals that have frequencies that are close to each other (say 60 and 61 Hz), you get a signal at roughly that same frequency (60 ish - technically 60.5, see discussion below) whose amplitude varies slowly, such as in the figure below. The slow oscillations are called “beats” – they occur when you add two signals of comparable amplitude and frequency. The result is like a "carrier wave" at the original frequency, which is modulated at a much lower frequency (like AM radio). It is easy to find the beat frequency (the slow frequency of modulation). Assume you add two signals of equal amplitude Vo, at frequencies f1 and f2 where f1 ≈ f2. The average frequency is ƒ = (f1 + f2)/2 and the difference is Af = f2 - f1. Reversing this, f₁ = ƒ – ▲ƒ/2 and ƒ₂ = ƒ + Aƒ/2. Written in terms of angular frequencies (w 2πf) the sum of signals is: = VSUM = Vo{sin w₁t + sin w₂t} = Vo{sin [(w - Aw/2)t] + sin [(w + Aw/2)t]} where is the average radian frequency. (a) Use the trig identity sin (a + b) = sin (a) cos (b) ± sin (b) cos (a) on each term in VSUM and reduce your expression to a product of a "fast" term (dependent on w) and a "slow" term (dependent on Aw). The slow term "modulates" (controls the amplitude of) the fast term. (b) What is the frequency of modulation ("beat frequency"), in terms of f1 and f₂? (c) What would a low pass filter with a cutoff frequency of ƒ /2 do to the signal?
Summing two frequencies: When two signals that have frequencies that are close to each other (say 60 and 61 Hz), you get a signal at roughly that same frequency (60 ish - technically 60.5, see discussion below) whose amplitude varies slowly, such as in the figure below. The slow oscillations are called “beats” – they occur when you add two signals of comparable amplitude and frequency. The result is like a "carrier wave" at the original frequency, which is modulated at a much lower frequency (like AM radio). It is easy to find the beat frequency (the slow frequency of modulation). Assume you add two signals of equal amplitude Vo, at frequencies f1 and f2 where f1 ≈ f2. The average frequency is ƒ = (f1 + f2)/2 and the difference is Af = f2 - f1. Reversing this, f₁ = ƒ – ▲ƒ/2 and ƒ₂ = ƒ + Aƒ/2. Written in terms of angular frequencies (w 2πf) the sum of signals is: = VSUM = Vo{sin w₁t + sin w₂t} = Vo{sin [(w - Aw/2)t] + sin [(w + Aw/2)t]} where is the average radian frequency. (a) Use the trig identity sin (a + b) = sin (a) cos (b) ± sin (b) cos (a) on each term in VSUM and reduce your expression to a product of a "fast" term (dependent on w) and a "slow" term (dependent on Aw). The slow term "modulates" (controls the amplitude of) the fast term. (b) What is the frequency of modulation ("beat frequency"), in terms of f1 and f₂? (c) What would a low pass filter with a cutoff frequency of ƒ /2 do to the signal?
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Chapter17: Sound
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![Summing two frequencies:
When two signals that have frequencies that are close to each other (say 60 and 61 Hz), you
get a signal at roughly that same frequency (60 ish - technically 60.5, see discussion below)
whose amplitude varies slowly, such as in the figure below.
The slow oscillations are called “beats” – they occur when you add two signals of
comparable amplitude and frequency. The result is like a "carrier wave" at the original
frequency, which is modulated at a much lower frequency (like AM radio). It is easy to find
the beat frequency (the slow frequency of modulation). Assume you add two signals of
equal amplitude Vo, at frequencies f1 and f2 where f1 ≈ f2. The average frequency is
ƒ = (f1 + f2)/2 and the difference is Af = f2 - f1. Reversing this, f₁ = ƒ – ▲ƒ/2
and ƒ₂ = ƒ + Aƒ/2. Written in terms of angular frequencies (w 2πf) the sum of
signals is:
=
VSUM = Vo{sin w₁t + sin w₂t} = Vo{sin [(w - Aw/2)t] + sin [(w + Aw/2)t]}
where is the average radian frequency.
(a) Use the trig identity sin (a + b) = sin (a) cos (b) ± sin (b) cos (a) on each term in
VSUM and reduce your expression to a product of a "fast" term (dependent on w) and a
"slow" term (dependent on Aw). The slow term "modulates" (controls the amplitude of) the
fast term.
(b) What is the frequency of modulation ("beat frequency"), in terms of f1 and f₂?
(c) What would a low pass filter with a cutoff frequency of ƒ /2 do to the signal?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf700b2b-ee54-42e6-9a3e-c60c572fd501%2F9d7d097f-8f93-46f3-b056-36abb208d553%2Fx54vaqb_processed.png&w=3840&q=75)
Transcribed Image Text:Summing two frequencies:
When two signals that have frequencies that are close to each other (say 60 and 61 Hz), you
get a signal at roughly that same frequency (60 ish - technically 60.5, see discussion below)
whose amplitude varies slowly, such as in the figure below.
The slow oscillations are called “beats” – they occur when you add two signals of
comparable amplitude and frequency. The result is like a "carrier wave" at the original
frequency, which is modulated at a much lower frequency (like AM radio). It is easy to find
the beat frequency (the slow frequency of modulation). Assume you add two signals of
equal amplitude Vo, at frequencies f1 and f2 where f1 ≈ f2. The average frequency is
ƒ = (f1 + f2)/2 and the difference is Af = f2 - f1. Reversing this, f₁ = ƒ – ▲ƒ/2
and ƒ₂ = ƒ + Aƒ/2. Written in terms of angular frequencies (w 2πf) the sum of
signals is:
=
VSUM = Vo{sin w₁t + sin w₂t} = Vo{sin [(w - Aw/2)t] + sin [(w + Aw/2)t]}
where is the average radian frequency.
(a) Use the trig identity sin (a + b) = sin (a) cos (b) ± sin (b) cos (a) on each term in
VSUM and reduce your expression to a product of a "fast" term (dependent on w) and a
"slow" term (dependent on Aw). The slow term "modulates" (controls the amplitude of) the
fast term.
(b) What is the frequency of modulation ("beat frequency"), in terms of f1 and f₂?
(c) What would a low pass filter with a cutoff frequency of ƒ /2 do to the signal?
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