# Modulation Essay

6411 words - 26 pages

Chapter 5 Traditional Analog Modulation Techniques
Mikael Olofsson — 2002–2007 Modulation techniques are mainly used to transmit information in a given frequency band. The reason for that may be that the channel is band-limited, or that we are assigned a certain frequency band and frequencies outside that band is supposed to be used by others. Therefore, we are interested in the spectral properties of various modulation techniques. The modulation techniques described here have a long history in radio applications. The information to be transmitted is normally an analog so called baseband signal. By that we understand a signal with the main part of its spectrum around zero. Especially, that ...view middle of the document...

In Chapter 3, Theorem 9, we noted that a convolution in the time domain corresonds to a multiplication in the frequency domain. In fact, the opposite is also true.

Theorem 10 (Fourier transform of a multiplication) Let a(t) and b(t) be signals with Fourier transforms A(f ) and B(f ). Then we have F {a(t)b(t)} = (A ∗ B)(f ).

Proof: The proof is along the same line as the proof of Theorem 8, but starting with the inverse transform of the suggested spectrum. Based on our deﬁnitions, we have
∞ ∞ ∞ j2πf t

F

−1

{(A ∗ B)(f )} =
−∞

(A ∗ B)(f )e

df =
−∞ −∞

A(φ)B(f − φ) dφ ej2πf t df.

We can rewrite the expression above as
∞ ∞

F

−1

{(A ∗ B)(f )} =
−∞ −∞

A(φ)B(f − φ)ej2πf t dφ df.

Now, set λ = f − φ, and we get
∞ ∞ ∞ ∞

F −1 {(A ∗ B)(f )} =
−∞ −∞

A(φ)B(λ)ej2π(λ+φ)t dτ dλ =
−∞

A(φ)ej2πφt dφ
−∞

B(λ)ej2πλt dλ.

Finally, we identify the last two integrals as the inverse Fourier transforms of A(f ) and B(f ), and we get F −1 {(A ∗ B)(f )} = a(t)b(t). 2 So, multiplying in the time domain corresponds to a convolution in the frequency domain.

5.1. Amplitude Modulation
3 2 1 0 −1 −2 −3 −1 0 1 antenna diode BP ﬁlter LP ﬁlter 1 0 −1 earphone −2 −3 −1 0 3 2

69

1

(a)

(b)

(c)

Figure 5.1: (a) A standard AM signal for the message m(t) = cos(2πt) with C = 2 and A = 1. The dark line is C + m(t). (b) Principle of an envelope detector. (c) The corresponding output from an envelope detector.

Standard AM
An AM signal, x(t), corresponding to the message signal, m(t), is given by the equation x(t) = A(C + m(t)) cos (2πfc t) , where fc is referred to as the carrier frequency, A is some non-zero constant, and where the constant C is chosen such that |m(t)| < C holds for all t. In Figure 5.1a a standard AM signal is presented together with the message, which in this particular example is a cosine signal. We mentioned that AM signals can be detected using a nonlinearity. The ﬁrst AM receiver was the so called crystal receiver. It consists of an antenna, a resonance circuit (bandpass ﬁlter), a diode and a simple low-pass ﬁlter. It extracts the envelope C + m(t) from x(t), and is therefore often called an envelope detector. The diode in Figure 5.1b is the nonlinearity that makes the detection possible. The few simple components makes it possible to manufacture the receiver at a low cost. In addition to that, it doesn’t even need a power source of its own. The power is taken directly from the antenna. The output power is of course very small, and only one listener could use the small earphone that was used. In Figure 5.1c, the AM signal is presented together with the output of an envelope detector. Note that the output is very similar to the original message. The mechanical parts in the earphone, and the ear will further low-pass ﬁlter the output, so the listener will hear almost the same signal as the one transmitted. Modern envelope detectors have ampliﬁers in various places and may be...

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