**Representation of DSB-AM signal: **

Mathematically, a DSB-AM wave can be represented as follows:

Let the carrier signal is denoted by c(t)=A_{c}sin(2πf_{c}t) , and the modulating signal is denoted by m(t).

The modulated signal C_{m}(t) can be written as C_{m}(t)=A_{c}(1+m(t))sin(2πf_{c}t).

**Spectrum of modulated signal:**

If M(f) represents the Fourier transform of the modulating signa m(t) ,then the spectrum of the amplitude modulated signal C_{m}(t) can be written as

C_{m}(f)=A_{c}/2j[δ(f-f_{c})-δ(f+f_{c})]+A_{c}/2j[M(f-f_{c})-M(f+f_{c})]

**Single tone modulation:**

Let us consider a special case when the modulating signal m(t) is given by m(t)=A_{m}sin(2πf_{m}t) and *f*_{m}<_{c}

The modulated waveform can be represented as:

C_{m}(t)=(A_{c}+A_{m}sin(2πf_{m}t))sin(2πf_{c}t)=A_{c}(1+msin(2πf_{m}t))sin(2πf_{c}t)

Here,m is the index of modulation and m≤l is usually used.

The modulated waveform for typical values m=0.5, Ac=2v,fm=1Khz and fm=10Khz is shown in the figure1 below.

Figure-1: Amplitude modulated wave

If m>1, the signal is over modulated and for such cases the demodulated waveform will be distorted. Figure 2 shows an over-modulated AM wave.

Fig.2 Over-modulated wave

C_{m}(t) can be written as

C_{m}(t)=A_{c}sin(2πf_{c}t)+A_{c}msin(2πf_{m}t)sin(2πf_{c}t)

= A_{c}sin(2πf_{c}t)+A_{c}m/2 cos (2π(f_{c}- f_{m})t)-A_{c}m/2 cos(2π(f_{c}+f_{m})t)

Thus we find that for a single tone AM, the modulated signal has three frequency components, namely, f_{c},(f_{c}+f_{m}) and (f_{c}- f_{m}).

Since C_{m}(t)=(A_{c}+A_{m}sin(2πf_{m}t))sin(2πf_{c}t)=A_{c}(1+msin(2πf_{m}t))sin(2πf_{c}t), the envelope of the carrier has the wave shape of the modulating signal.

For the Figure 3, A_{max}=A_{c}(1+m) and A_{min}=A_{c}(1-m) respectively

Figure-3: Envelope variation of amplitude modulated carrier

Therefore we can write,

Given the sinusoidally modulated wave as shown in Figure 3, the modulation index m can be computed by measuring A_{max} and A_{min}

The power in the carrier wave component is given by

The power in each of the side frequencies is given by

The total power P _{T} is given by

With m=1, only 25% of the carrier power is present at the side frequencies.

The merit of amplitude modulated carrier is the ease with the baseband signal can be recovered. The process of recovery of modulating signal is called demodulation. The demodulation of DSB AM signal can be done either by using an envelope detector or by passing the amplitude modulated signal through a non-linear device. In this experiment, we shall consider demodulation using envelope detector.

Details of generation of AM signals and operation of envelope detector will be covered in the section where experimental setups will be discussed.