MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science : Signals and Systems Spring 2007

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems Spring 2007 Tutorial 9: Week of April 6 Announcements Due to holidays on April 6 and 7, no tutorials will be held this week. A Quiz 2 Review will be held on Wednesday, Aril 8 from 7:30-9:30pm at 34-0, and Thursday, April 9 from 7:30-9:30 pm in In the week of April 23, the TAs will jointly hold office hours/open tutorials from 0 am - 6 pm on Monday, April 23 and again from 0 am - 6 pm on Tuesday, April 24 in the TA offices. ] Quiz 2 is on Tuesday April 24 from 7:30-9:30pm. The materials in this tutorial will not be covered in Quiz 2. However, they will be surely covered on the Final. Topics covered in this note: AM Modulation Asynchronous Demodulation Double-Sideband (DSB) and Single-Sideband (AM) Frequency-Division Multiplexing (FDM) The Superheterodyne Receiver AM Modulation with an Arbitrary Periodic Carrier Time-Division Multiplexing (FDM) Sinusoidal Frequency Modulation (FM) Discrete-Time Sinusoidal Amplitude Modulation

2 AM Modulation AM Modulation is used to transmit an information signal at some desired frequency range. It is typically more efficient to transmit signals at higher frequencies. The information or modulating signal must then be shifted to the desired higher frequency. Amplitude modulation involves multiplying a carrier signal c(t) by the magnitude of the information signal. The modulated signal = c(t). One method of AM modulation uses a complex exponential carrier signal c(t) = e jct. Multiplication of the information signal by an exponential shifts its spectrum up by an amount equal to the carrier frequency c. The input signal can then be recovered from the modulated signal by multiplication of the modulated signal by the conjugate of the exponential carrier e jct which shifts the spectrum back. Another method used for amplitude modulation involves multiplication of the modulating signal by a sinusoidal carrier (cosine) c(t) = cos( c t) and therefore the modulated signal = cos( c t). This replicates that spectrum of the original signal around ± c and scales by 2. This procedure works if the carrier frequency c is greater than the highest frequency of the input signal M. Demodulation is achieved by modulating with the same sinusoidal carrier. This creates a copy back at the baseband as well as replicates of the original spectrum at ±2 c. These high frequency replicates can then be filtered out using a lowpass filter. For synchronous demodulation, the exponential or cosine used for demodulation has to be in phase with the carrier signal used for modulation, otherwise the recovered signal will be attenuated depending on the discrepancy on phase and possibly vanish. AM Modulation with Exponential Carrier c(t) = e jct e jct AM Modulation with Sinusoidal Carrier w(t) Ideal lowpass filter cos( c t) cos( c t) 2 Asynchronous demodulation Asynchronous demodulation can be employed to avoid having to synchronize the modulator and demodulator. If the modulating signal is positive and the carrier frequency c is much 2

3 higher than the M, the highest frequency in the modulating signal, then can be recovered using an envelope detector. If is not positive everywhere, the same sinusoidal carrier with a sufficiently large carrier amplitude A is added to the modulating signal = (A + )cos( c t). The amplitude A must be greater than the maximum value of. Envelope detection is then performed on the modulated signal. Envelope Detector A cos( c t) 3 Double-Sideband (DSB) and Single-Sideband (SSB) AM Double sideband modulation involves multiplication of the modulating signal by a carrier cosine and keeping both upper and lower sidebands. Since the modulated signal occupies twice the bandwidth of the original, the use of bandwidth is inefficient. Single-sideband modulation (SSB) is achieved if only the upper or lower sidebands are retained. Since the modulating signal is real, all the information needed is available in the upper or lower sidebands. This makes more efficient use of the bandwidth. The disadvantage is added complexity to the modulator. 4 Frequency Division Multiplexing (FDM) Given a communication channel with a wide bandwidth, several signals which are overlapping in frequency can be sent simultaneously by having their frequency content shifted to different bands by sinusoidal amplitude modulation so that the spectra of the modulated signals no longer overlap. This is achieved by using different frequencies (spaced apart sufficiently) for the carrier sinusoids (cosines). The multiplexed signals occupy distinct segments of the frequency band. The individual signals are assumed to be bandlimited. Demultiplexing is achieved for each of the channels by bandpass filtering, followed by demodulation. Demodulation involves multiplication by a carrier sinusoid at the appropriate frequency to create a copy in the baseband and lowpass filtering to remove higher frequency replicates. 3

4 Frequency Division Multiplexing System and Demodulation to Recover x a (t) cos( a t) x a (t) cos( b t) cos( a t) x b (t) w(t) cos( b t) w(t) Bandpass Filter Lowpass Filter x a (t) x c (t) 5 Superheterodyne Receiver Demultiplexing in FDM requires a sharp cutoff bandpass filter with a variable center frequency. Since variable frequency-selective filters are difficult to implement, a fixed filter is implemented instead, and an intermediate stage of modulation and filtering is used. Instead of tuning a sharp cutoff filter to select the modulated signal, it is the signal that is shifted toward a sharp cutoff fixed bandpass filter located at ± IF. The shifting from ± a to IF is achieved by modulating a cos(( a + IF )t) carrier. The coarse tunable filter is centered at the desired spectrum to be demodulated, ± a. It is used to filter out spectral content at ±( a + 2 IF ) which will otherwise also get shifted to ± IF. Superheterodyne Receiver to Recover x a (t) cos(( a + IF )t) w(t) Coarse Tunable Sharp Fixed x a (t) Bandpass Filter Bandpass Filter 6 AM Modulation with an Arbitrary Periodic Carrier Modulation can also be achieved using an arbitrary periodic pulse train as the carrier. Equally spaced time slices of are then transmitted. The modulated signal is the sum of scaled and 4

5 shifted replicas of the input spectrum X(j) where the scaling factors correspond to the Fourier series coefficients of the periodic pulse train c(t). Demodulation is achieved by lowpass filtering. It is possible to recover as long as the frequency of the pulse train c = 2π T is at least twice the highest frequency of the input, M, and the DC Fourier coefficient of the pulse train is nonzero. Ideal lowpass filter c(t) = k= a ke jkct c(t) 0 T 2T t 7 Time-Division Multiplexing Amplitude modulation with a pulse-train carrier can be used to transmit several signals over a single channel. Each signal is multiplied by a pulse train which corresponds to the signal s alloted time slots. The time slots are spaced T apart and have duration. Time-shifted versions of the same pulse train are multiplied by the other signals. The process works as long as there is no overlapping between the time slots of different signals. 8 Sinusoidal Frequency Modulation This modulation technique uses the modulating signal to control the frequency of the sinusoidal carrier c(t). Since the envelope of the carrier is constant, an FM transmitter can always operate at peak power. An advantage of this is that amplitude variations introduced over a transmission channel due to additive disturbances or fading can be eliminated at the receiver. The frequency of the modulated signal is offset from the carrier frequency by an amount proportional to the amplitude of the modulating signal. The modulated signal is expressed as = A cos(θ(t)) where dθ(t) dt = c + k f. 9 Discrete-Time Sinusoidal Amplitude Modulation This procedure is analogous to the continuous time modulation achieved by multiplication by an exponential or a sinusoid. Multiplication by an exponential c[n] = e jcn causes a shift of the periodic spectrum in frequency. Multiplication by the conjugate of the carrier exponential shifts the 5

6 spectrum back and achieves demodulation. The other alternative is multiplication by a sinusoidal carrier c[n] = cos[ c n]. The only difference in the spectral analysis is that since the spectrum of the modulating signal X(e j ) is periodic and a periodic convolution is performed, an additional condition on the carrier frequency is imposed such that there is no overlapping/aliasing. In this case, the constraint on the carrier frequency is M < c < π M. AM Modulation with Exponential Carrier x[n] y[n] y[n] x[n] c[n] = e jcn e jcn AM Modulation with Sinusoidal Carrier x[n] y[n] y[n] w[n] Ideal lowpass filter x[n] cos( c n) cos( c n) 0 Problems Problem 9. (O&W Problem 8.36) Problem 9.2 (O&W Problem 8.37) Problem 9.3 (O&W Problem 8.48) 6

7 Problem 9.4 Assume we are given two signals a(t) and b(t) whose Fourier transforms are given by A(j) B(j) m m m m For each of the two systems in Part a and Part b, determine whether the following conditions are true c(t) is proportional to a(t), i.e., c(t) = K a(t) for some constant K. d(t) is proportional to b(t), i.e., d(t) = K 2 b(t) for some constant K 2. System For each system, provide an explanation to justify your answer. a(t) H(j) c(t) p(t)=cos( t) b(t) H(j) d(t) p(t)=cos( t) H(j) = for < 0 for > System 2 a(t) G(j) j H(j) c(t) -j r(t)=cos(2 t) p(t)=sin(2 t) b(t) G(j) j -j H(j) d(t) G(j) = -j for > 0 j for < 0 H(j) = for < 0 for > 7