1 EEE 304 Lab Exercise 4: 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 electro-magnetic 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, the signal in a cell phone is a voice signal with a bandwidth of about 4 KHz. The typical frequency of the transmitted and received signal is several hundreds of megahertz to a few gigahertz. 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 ωc is varied in proportion to baseband signal m(t). In the remainder of this lab exercise we will interchangeably use the notations (ω and f) to represent frequency. f denotes the frequency of a sinusoidal signal in Hz, whereas ω 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) = 𝑨𝒄𝒐𝒔(𝝎𝒄𝒕 + 𝜽𝒄), is varied in proportion to the amplitude of the modulating (message) signal m(t). The frequency and the phase of the carrier are fixed. 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(𝝎𝒄𝒕) . The carrier signal 𝑨𝒄𝒐𝒔(𝝎𝒄𝒕) is added to this signal, and the resulting signal 𝚽 𝑨𝑴(𝒕) = (𝑨 + 𝒎(𝒕))𝐜𝐨𝐬(𝝎𝒄𝒕) is referred as the amplitude modulated signal. 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. 2 Figure 1. Illustration of Amplitude Modulation. Let the Fourier transform of the message signal be denoted by M(ω), i.e., F[m(t)] = M(ω) (1) From properties of Fourier Transform, it can be easily observed that F [m(t)cos(ωct)] = [M(ω+ωc) + M(ω-ωc)] / 2 (2) Hence, the modulated signal and its Fourier transform are given by Φ 𝐴𝑀(𝑡) = (𝐴 + 𝑚(𝑡))cos(𝜔𝑐𝑡) (3) F [Φ𝐴𝑀(𝑡)] = [M(ω+ωc)+M(ω-ωc)]/2 + πA[𝛿(ω+ωc)+𝛿(ω-ωc)] (4) Recall that M(ω-ωc) is M(ω) shifted to the right by ωc and M(ω+ωc) is M(ω) shifted to the left by ωc. The process of modulation in time and frequency domains is described in Figure 2 and 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 reduced from 2A to A. Figure 2. Modulating signal m(t) in time (left) and frequency (right). Figure 3. Modulated signal in time (left) and frequency (right). 3 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. Here, we describe coherent demodulation, as it is very similar to the modulation process and easy to demonstrate and understand. 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 signal is Φ𝐴𝑀(𝑡) = (𝐴 + 𝑚(𝑡))cos(𝜔𝑐𝑡). Multiplying it with cos(ωct) results in the signal r(t), where r(t) is given by r(t) = (A+m(t)) cos2(ωct) = (A+m(t)) (1 + cos 2ωct)/2. (5) Its Fourier transform is 𝑅(𝜔) = 𝐴 2 𝛿(𝜔) + 𝜋𝐴 2 [𝛿(𝜔 + 2𝜔𝑐) + 𝛿(𝜔 − 2𝜔𝑐)] + 1 2 𝑀(𝜔) + 1 4 [𝑀(𝜔 + 2𝜔𝑐) + 𝑀(𝜔 − 2𝜔𝑐)] (6) It is evident from the equation above that 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(ω), 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. 3. Modulation Index Modulation index (µ) is defined as the ratio between the amplitude 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 4 erroneous signal reconstruction if non-coherent demodulation (envelope detection) is used, but in case of coherent demodulation, any value of µ provides reconstruction. The following figure shows modulated signals with different modulation index value. The carrier signal has amplitude of 1. The baseband signal has amplitude of 0.5, 1 and 1.5. All baseband signals are shifted up by 1 (amplitude of the carrier signal). Figure 6. Modulated signal with different modulation index1 1 https://en.wikipedia.org/wiki/Amplitude_modulation#Modulation_index