DC
UNIT - 2Source Coding Q1) The continuous-time signal x(t) = cos(200πt) is used as the input for a CD converter with the sampling period 1/300 sec. Determine the resultant discrete-time signal x[n]. S2)
Q2) Consider an analog signal.
S2)The frequency in the analog signal
The largest frequency is
The Nyquist rate is
Q3) The analog signal
What is the Nyquist rate for this signal? Using a sampling rate 
. What is discrete time signal obtained after sampling? What is analog signal 
we can reconstruct from the samples if we use ideal interpolation? S3)
Q4) 
Find the minimum sampling rate required to avoid aliasing. If 
, what is the discrete time signal after sampling? If 
, what is the discrete time signal after sampling? What is the frequency F of a sinusoidal that yields sampling identical to obtained in part c? S4)
Q5) Suppose a continuous-time signal x(t) = cos (Ø0t) is sampled at a sampling frequency of 1000Hz to produce x[n]: x[n] = cos(πn/4) Determine 2 possible positive values of Ø0, say, Ø1 and Ø2. Discuss if cos(Ø1t) or cos(Ø2t) will be obtained when passing through the DC converter.
Q6) Explain PCM.A6) In Pulse Code Modulation, the message signal is represented by a sequence of coded pulses. This message signal is achieved by representing the signal in discrete form in both time and amplitude.
We know, X[n] =x(nT) = cos(200πnT) = cos(2πn/3) , where n= -1,0,1,2……
The frequency in x(t) is 200π rad/s while that of x[n] is 2π/3.
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S2)The frequency in the analog signalThe largest frequency isThe Nyquist rate is Q3) The analog signal . What is discrete time signal obtained after sampling? we can reconstruct from the samples if we use ideal interpolation?
2. For
For Hence
So that normalizing frequencies are
The analog signal that we can recover is
Which is different than the original signal
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, what is the discrete time signal after sampling? , what is the discrete time signal after sampling? a. The minimum sampling rate is
And the discrete time signal is
b. if
c. If Fs=75Hz , the discrete time signal is
d. For the sampling rate
So, the analog sinusoidal signal is
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S5) Taking T= 1/1000s cos(πn/4) =x[n] = x(nT) = cos (Ø0n/1000) Ø1 is easily computed as Ø1 = 250π
Ø2 can be obtained by noting the periodicity of a sinusoid:
Ø2 = 2250π
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Basic Elements of PCMThe transmitter section comprises of Sampling, Quantizing and Encoding. The low pass filter prior to sampling prevents aliasing of the message signal.The basic operations in the receiver section are regeneration of impaired signals, decoding, and reconstruction of the quantized pulse train.
Fig.: PCM
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Low Pass FilterThis filter eliminates the high frequency components present in the input analog signal to avoid aliasing of the message signal.
SamplerIt helps to collect the sample data at instantaneous values of message signal, so as to reconstruct the original signal. The sampling rate must be in accordance with the sampling theorem.
QuantizerIt reduces excessive bits and confines the data. It reduces the redundant bits and compresses the value of the sampled output.
EncoderThe digitization of analog signal is done by the encoder. It designates each quantized level by a binary code. Encoding minimizes the bandwidth used.
Regenerative RepeaterIt increases the signal strength. The output of the channel also has one regenerative repeater circuit, to compensate the signal loss and reconstruct the signal, and also to increase its strength.
DecoderThe decoder circuit decodes the pulse coded waveform to reproduce the original signal. This circuit acts as the demodulator.
Reconstruction FilterAfter the digital-to-analog conversion is done by the regenerative circuit and the decoder, a low-pass filter is employed, called as the reconstruction filter to get back the original signal.Hence, the Pulse Code Modulator circuit digitizes the given analog signal, codes it and samples it, and then transmits it in an analog form. This whole process is repeated in a reverse pattern to obtain the original signal. Q7) What is DPCM? Explain.S7) Differential pulse code modulation (DPCM) is a procedure of converting an analog into a digital signal in which an analog signal is sampled and then the difference between the actual sample value and its predicted value (predicted value is based on previous sample or samples) is quantized and then encoded forming a digital value. DPCM code words represent differences between samples unlike PCM where code words represented a sample value. Basic concept of DPCM - coding a difference, is based on the fact that most source signals show significant correlation between successive samples so encoding uses redundancy in sample values which implies lower bit rate. Realization of basic concept (described above) is based on a technique in which we have to predict current sample value based upon previous samples (or sample) and we have to encode the difference between actual value of sample and predicted value (the difference between samples can be interpreted as prediction error).
Because it's necessary to predict sample value DPCM is form of predictive coding.DPCM compression depends on the prediction technique, well-conducted prediction techniques lead to good compression rates, in other cases DPCM could mean expansion comparing to regular PCM encoding.
Fig.: DPCM Q8) What is delta modulation? Write features of delta modulation.S8) The type of modulation, where the sampling rate is much higher and in which the step size after quantization is of a smaller value Δ, such a modulation is termed as delta modulation.
Because it's necessary to predict sample value DPCM is form of predictive coding.DPCM compression depends on the prediction technique, well-conducted prediction techniques lead to good compression rates, in other cases DPCM could mean expansion comparing to regular PCM encoding.
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Features The quality is moderate. The design of the modulator and the demodulator is simple. The stair-case approximation of output waveform. The step-size is very small, i.e., Δ delta. The bit rate can be decided by the user. This involves simpler implementation. Delta Modulation is a simplified form of DPCM technique, also viewed as 1-bit DPCM scheme. As the sampling interval is reduced, the signal correlation will be higher. Q9) Explain ADM in detail.A9) In digital modulation, we have come across certain problem of determining the step-size, which influences the quality of the output wave.A larger step-size is needed in the steep slope of modulating signal and a smaller step size is needed where the message has a small slope. The minute details get missed in the process. So, it would be better if we can control the adjustment of step-size, according to our requirement in order to obtain the sampling in a desired fashion. This is the concept of Adaptive Delta Modulation.Following is the block diagram of Adaptive delta modulator.
Fig. : Adaptive delta modulator The gain of the voltage-controlled amplifier is adjusted by the output signal from the sampler. The amplifier gain determines the step-size and both are proportional.ADM quantizes the difference between the value of the current sample and the predicted value of the next sample. It uses a variable step height to predict the next values, for the faithful reproduction of the fast-varying values. Q10) Determine the Nyquist frequency and Nyquist rate for the continuous-time signal x(t) which has the form of:X(t) = 1+ sin(2000πt) + cos (4000πt) S10) The frequencies are 0, 2000π and 4000π. The Nyquist frequency is 4000π rad/s and the Nyquist rate is 8000π rad/s.
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