EEE203 Signals and Systems I

Tutorial: Amplitude Modulation

Modulation is the process of varying one or more properties of a periodic waveform, called the carrier signal (typically of very high frequency), with a modulating signal that generally contains information to be transmitted. There are two motivating reasons for modulation:

1)    Modulation allows for the use of small antennae in message transmission therefore making the application portable e.g. mobile phones.

2)    It also allows us to multiplex, or share, a communication medium among many concurrently active users through the choice of different carrier frequencies separated by a frequency gap band. This technique is known as Frequency Division Multiplexing.

Radio, television, GPS, mobile phones and all other wireless communications devices transmit information across  distances  using  electromagnetic  waves.  To  send these waves  across  long distances in free space, the frequency of the transmitted signal must be quite high compared to the frequency of the information signal. This keeps aliasing at bay as well as help keep antenna sizes small. For example, a voice signal has a bandwidth of about 4 KHz. The typical frequency of the transmitted and received signal is several hundreds of megahertz to a few gigahertz.

Antenna size is in general proportional to wavelength of the transmitted electromagnetic waves. Let us look at how the antenna size can be made smaller with higher carrier frequencies. For example, the wavelength of a 1 GHz electromagnetic wave in free space is 30 cm, whereas a 1 kHz electromagnetic wave is one million times larger, 300 km, it would be impossible to build and power such a behemoth!

Communication that uses modulation to shift the frequency spectrum of a signal is known as  carrier communication. In this mode, one of the basic parameters (amplitude, frequency, or phase) of a sinusoidal carrier of high frequency uc is varied in proportion to baseband signal m(t). We will  focus on amplitude modulation in this tutorial. We will interchangeably use the notations (u and f) to represent frequency. f denotes the frequency of a sinusoidal signal in Hz, whereas u is the  frequency in rad/sec.

1. Amplitude Modulation

Amplitude modulation is characterized by the fact that the amplitude A of the carrier signal c (t) is varied in proportion to the amplitude of the modulating (message) signal m(t). The frequency and the phase of the carrier are fixed. The carrier signal c (t) is represented as

c(t) = Acos(wct + θc).                                                                 (1)

For simplicity, we can assume that θc  = 0 in the carrier signal. If the carrier amplitude A is made directly proportional to the modulating signal m(t), we obtain the signal m(t)cos (wct). When

the carrier signal Acos(w ct) is added to this signal, the resulting amplitude modulated signal φAM (t) is represented as:

φAM (t) = (A + m(t))cos(wct).                                                 (2)

Figure 1 illustrates the amplitude modulation. The modulating (message) signal is multiplied by

the carrier signal. The modulating signal forms the envelope of the modulated signal.

Figure 1. Illustration of Amplitude Modulation.

Let the Fourier transform. of the modulating signal be denoted by M(ju), i.e.,

F

  m(t) ↔ M(jw)                                                                                    (3)

The time and frequency domain plots of the modulating signal is shown in Figure 2.

Figure 2. Modulating signal m(t) in time (left) and frequency (right).

What is the Fourier transform. of the amplitude modulated signal ΦAM(jw)? You can get a hint from the time and frequency domain plots of the modulated signal from Figure 3.

Figure 3. Modulated signal in time (left) and frequency (right).

As shown in Figure 3, there is a replication of the spectrum of the modulating signal, each centered around -ωc and ωc. A portion that lies above the ωc is called Upper Side Band (USB) and a portion lies below ωc is called Lower Side Band (LSB). For each replication the amplitude is half of the original height, i.e., reduced from 2α to α . Why are there impulses at -ωc and ωc?

2. Amplitude Demodulation

Demodulation can be performed in two different ways.

●    Coherent Demodulation: This assumes that the carrier signal with same frequency and phase can be generated at the receiver for demodulation.

●    Non-Coherent Demodulation: This directly demodulates the received signal from its

envelope without requiring the carrier signal. The envelope detector can be implemented very inexpensively using a half wave rectifier followed by a low pass filter.

Here, we will describe coherent demodulation, as it is very similar to the modulation process and easy to demonstrate and understand. Recall the scheme of modulation shifts the spectrum  of the message signal by multiplying it with the carrier signal. Hence, demodulation requires the shifting of the spectrum back to the original location. To achieve this, we multiply again the modulated signal with the carrier signal, as described in Figure 4 and Figure 5.

Figure 4. Demodulation.

Figure 5. Frequency spectrum of modulated signal multiplied by the carrier. The dashed line indicates the frequency response of a low pass filter used to extract the demodulated signal.

The received modulated signal is φAM (t) =  (A + m(t))cos(w ct). Multiplying it with cos(w ct) results in the signal r(t) given by

r(t) = (A + m(t))cos2 (w ct) = 2/1 (A + m(t))(1 + cos(2w ct))                    (4)

What is the Fourier transform. R(jω) of r(t)?  Again you can get a hint from Figure 5.

As shown in Figure 5, one part of the demodulated signal spectrum is centered at zero frequency and the other part is centered at ±2ωc. In order to retrieve the original message signal M(jω), and to remove the high frequency components, r(t) is passed through a low pass filter with a cutoff frequency greater than the bandwidth B Hz of the message signal. What about the impulse at zero frequency? How to remove it?

3.    Modulation Index

Modulation index (µ) is defined as the ratio between the peak value of the message signal and the amplitude of the carrier signal. This indicates how much the modulated signal varies around its  original  level.  The  value  of  µ  < 1 results in under-modulation and µ >  1  results  in  over- modulation.   Over-modulation   results   in   erroneous   signal   reconstruction   if   non-coherent demodulation (envelope detection) is used, but in case of coherent demodulation, any value of µ provides reconstruction.

Figure 6 shows modulated signals with different modulation index value. The carrier signal has amplitude of 1. The modulating baseband signal has amplitude of 0.5, 1 and 1.5.

Figure 6. Modulated signal with modulation index µ = 0.5, µ = 1 and µ = 1.5

4.    Example

Suppose the modulating signal is a 100Hz cosine signal, i.e., m(t) = cos(2π100t). The signal and its Fourier transform (FFT, magnitude) are shown in Figure 7. In the frequency domain, you can see two impulses located at -100Hz and 100Hz, i.e., the frequency of the cosine signal.

Figure 7. Time domain and frequency domain plots of modulating signal

m(t) = cos(2π100t)

The modulating message signal m(t) is then modulated by a carrier signal c(t) = 2cos(2π1000t). The modulated signal φAM (t) =  (2 + m(t))cos(2π1000t) and its FFT are shown in Figure 8. In the time domain, the amplitude of the carrier signal is modulated by the message as shown by its envelope. The amplitude varies from 1 to 3 due to the 0.5 modulation index. In the frequency domain, the spectrum of the modulating signal is split into two, one half shifts to the left by 1000Hz, and the other half shifts to the right by 1000Hz. The impulses at - 1000Hz and 1000Hz are the FFT of the carrier signal c(t).

Figure 8. Time domain and frequency domain plots of modulated signal

φAM (t) = (2 + m(t))cos(2π1000t)

To demodulate the signal,  multiple  it  by cos(2π1000t),  i.e., r(t) = φAM (t) cos(2π1000t) = (2 + m(t))cos2 (2π1000t). The signal r(t) and its  FFT are shown in Figure 9. In the frequency domain, the spectrum of φAM (t) is split into two, one half shifts to the left by 1000Hz, and the other half shifts to the right by 1000Hz. Therefore, the cluster around -1000Hz in Figure 8 is shifted to -2000Hz and 0Hz, and the cluster around 1000Hz in Figure 8 is shifted to 0Hz and 2000Hz, as shown in Figure 9. Now if we pass r(t) through a  low pass filter to get rid of the two clusters around ±2000Hz, and remove the DC component (the impulse at 0Hz), and scale it properly, we can get the original signal m(t) back.

Figure 9. Time domain and frequency domain plots of demodulated signal

r(t) = φAM (t) cos(2π1000t) = (2 + m(t))cos2 (2π1000t)

One last note, the amplitude modulation scheme discussed in this tutorial is often referred to as “double sideband, full carrier” amplitude modulation. There are variations of this basic scheme.  Refer to this article for a quick introduction.