# Frequency spectrum of the modulated signal g(t)

• raymond23
You can use the expressions involving sin and cos of sums to rewrite the equation you just wrote, in terms of sin and cos of single-frequency terms. The final answer will involve just sin and cos of single-frequency terms, and no more sin or cos of sums.g(t) = sin(2πfct) + cos(2πfst+π3) sin(2πfct)g(t) = sin(2πfct) + cos(2πfst) sin(2πfct) cos(π3) + sin(2πfst) sin(2πfct) sin(π3)Now, you can use sin(x) cos(y) = (sin(x+y) +
raymond23

## Homework Statement

Let the baseband signal be
s(t)=cos(2πfst+π3), where fs=5kHz. Radio carrier is
c(t)=sin(2πfct), where fc=100MHz. Using the amplitude modulation of g(t)=(1+s(t))c(t), what is the frequency spectrum of the modulated signal g(t)?
What are the amplitude and phase shift of each frequency component in g(t)?

## Homework Equations

g(t)=(1+s(t))c(t)

## The Attempt at a Solution

g(t)= (1+ cos(2πfst+π3)) sin(2πfct)
g(t)= sin(2πfct)+ cos(2πfst+π3) sin(2πfct)
g(t)=

raymond23 said:
g(t)= sin(2πfct)+ cos(2πfst+π3) sin(2πfct)

Looks good so far. The first sin(...) term gives you one of the frequency components.

You'll need to express "cos(2πfst+π3) sin(2πfct)" as the sum of distinct, single-frequency sin and/or cos terms. You can do that using these trig identities:

sin(x + y) = sin(x)·cos(y) + cos(x)·sin(y)
sin(x - y) = sin(x)·cos(y) - cos(x)·sin(y)

p.s. welcome to PF

I try this
g(t)= sin(2πfct)+ cos(2πfst+π3) sin(2πfct)
g(t)= sin(2πfct)+ (cos(2πfst)+cos(π3)) sin(2πfct)
g(t)= sin(2πfct)+ (cos(2πfst)sin(2πfct)+cos(π3)sin(2πfct))
but don't know how to continue

raymond23 said:
I try this
g(t)= sin(2πfct)+ cos(2πfst+π3) sin(2πfct)
g(t)= sin(2πfct)+ (cos(2πfst)+cos(π3)) sin(2πfct)

There's a problem there, because
cos(2πfst+π3) and cos(2πfst)+cos(π3)​
are not equivalent.

If you add the two equations in my earlier post, you'll have an expression for
sin(x) cos(y)​
which will be useful here.

## 1. What is the frequency spectrum of a modulated signal?

The frequency spectrum of a modulated signal refers to the range of frequencies contained within the signal. It can be visualized as a graph with frequency on the x-axis and amplitude on the y-axis.

## 2. How is the frequency spectrum of a modulated signal different from an unmodulated signal?

An unmodulated signal typically has a single frequency, while a modulated signal contains multiple frequencies due to the addition of a carrier wave. This results in a more complex frequency spectrum.

## 3. How is the frequency spectrum affected by the modulation index?

The modulation index, which represents the amount of modulation in a signal, can affect the amplitude and distribution of frequencies in the frequency spectrum. A higher modulation index can result in a wider range of frequencies and a more prominent carrier signal.

## 4. Can the frequency spectrum be used to identify the type of modulation used?

Yes, the frequency spectrum can be used to identify the type of modulation used by analyzing the distribution of frequencies. For example, in amplitude modulation, the carrier frequency will be present along with sidebands at specific frequencies.

## 5. How can the frequency spectrum of a modulated signal be observed?

The frequency spectrum of a modulated signal can be observed using a spectrum analyzer, which displays the amplitude of different frequencies present in the signal. It can also be calculated using mathematical equations, depending on the type of modulation used.

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