Question

Consider a message signal with the spectrum

M(f)= tri(f/w)
The message bandwidth is 1500 Hz. This signal is applied to a product modulator, together with a carrier wave c(t) = 4 cos(2π fc t) producing the Double Sideband Suppressed Carrier (DSB-SC) modulated signal s(t).

a) (10 points) Determine an expression for the spectrum of DSB-SC modulated signal s(t).
b) (5 points) What is the lowest possible carrier frequency fc to avoid sideband overlap in the DSB-SC modulated signal s(t)?

c) (15 points) If the carrier frequency fc=5.5 kHz, then sketch the spectrum of s(t) and label it carefully by indicating important frequencies and amplitudes.

d) (15 points) If the carrier frequency fc=5.5 kHz, then suggest a demodulator that will recover the message signal m(t) without an amplitude gain or loss. Sketch the block diagram of the demodulator indicating important characteristics of used blocks (For example, if you are using a filter then indicate type, bandwidth, and gain of the filter).

Answer 1

a) The spectrum of the DSB-SC modulated signal s(t) can be obtained by multiplying the message signal M(f) with the carrier frequency c(t) as follows:

s(t) = M(f) * c(t) = tri(f/w) * 4 cos(2π fc t)

Using the trigonometric identity cos(A)cos(B) = 1/2 [cos(A+B)+cos(A-B)], we can write:

s(t) = 2tri(f/w)cos(2π fc t)cos(2π f t)

The spectrum of s(t) is therefore given by the product of the Fourier transforms of the triangular function and the cosine function:

S(f) = 2/2 [M(f-fc) + M(f+fc)] * 1/2 [δ(f-fc) + δ(f+fc)]

Simplifying, we get:

S(f) = tri((f-fc)/w) + tri((f+fc)/w)

b) The bandwidth of the DSB-SC modulated signal is twice the bandwidth of the message signal, i.e., 2 x 1500 = 3000 Hz. Therefore, the minimum carrier frequency to avoid sideband overlap is fc = 1500 Hz.

c) The spectrum of s(t) with fc = 5.5 kHz is shown below:

+-----------------------+

| |

| tri(f/w) |

| |

+------+-----------+-----------+-------+

-fc -1500 0 1500 3000 (Hz)

The important frequencies in the spectrum are:

Carrier frequency: fc = 5.5 kHz

Upper sideband frequency: fc + 1500 = 7.0 kHz

Lower sideband frequency: fc - 1500 = 4.0 kHz

d) A coherent demodulator can be used to recover the message signal without any amplitude gain or loss. The block diagram of the demodulator is shown below:

+--------------+

| |

| Local |

| Oscillator |

| cos(2π fct) |

| |

+-------+------+

|

|

v

+-------+------+

| |

| Product |

| Modulator |

| |

+-------+------+

|

|

v

+-------+------+

| |

| Low-pass |

| Filter |

| |

+--------------+

The received signal is multiplied with a local oscillator signal of the same frequency and phase as the carrier to obtain the product of the two signals. The resulting signal is then passed through a low-pass filter with cutoff frequency equal to the message bandwidth of 1500 Hz. The output of the filter is the demodulated message signal m(t).