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Unit-3

Digital Modulation

Q1) Explain Orthonormal Basis.

A1) Signal set {φk(t)}n is an orthogonal set if

If is an orthonormal set.

Consider a set of M signals (M-ary symbol){sit (t),i=1,2,…,M} with finite energy i.e.

Then, we can express each of these waveforms as weighted linear combination of orthonormal signals

where N ≤ M is the dimension of the signal space and are called the orthonormal basis functions

Let, for a convenient set of {ϕj (t)}, j = 1,2,…,N and 0 ≤ t <T,

such that

Now, we can represent a signal si(t) as a column vector whose elements are the scalar coefficient sj, j = 1, 2, ….., N :

These M energy signals or vectors can be viewed as a set of M points in an N – dimensional

Euclidean space, known as the Signal Space. Signal Constellation is the collection of M signals points (or messages) on the signal space

Fig.: Signal Constellation

Now the length or norm of a vector is denoted as . The squared norm is the inner product of the vector.

The cosine of the angle between )

Therefore, are orthogonal to each other if =0

If E is the energy of the i-th signal vector

forms an orthonormal set

For a pair of signals

It may now be guessed intuitively that we should use such that the Euclidean distance between them i.e. is as much as possible to ensure that their detection is more robust even in presence of noise. For example, if have same energy. E,(i.e. they are equidistance from the origin), then an obvious choice for maximum distance of separation is

Q2) Draw and explain coherent reception.

A2) Coherent reception is well known in wireless communication systems. In those wireless systems, a radio frequency (RF) local oscillator (LO) is tuned to “heterodyne”, which is a signal processing technique which combine a high-frequency signal f1 with another f2 to produce a lower frequency signal (f1 – f2), with a received signal through an RF mixer, as shown in Fig.(a), so that both the amplitude and phase information contained in an RF carrier can be recovered in the following digital signal processor (DSP).

For an optical coherent system, a narrow-linewidth tuneable laser, serving as an LO, tunes its frequency to “intradyne” with a received signal frequency through an optical coherent mixer, as shown in Fig. (b), and thereby recovers both the amplitude and phase information contained in a particular optical carrier.

Here, “intradyne” means that the frequency difference between an LO and a received optical carrier is small and within the bandwidth of the receiver, but does not have to be zero.

This implies that the frequency and phase of an LO do not have to be actively controlled to an extreme accuracy, therefore avoiding the use of a complicated optical phase locked loop.

Fig.: Coherent Receiver

In contrast to coherent detection is direct detection, typically used by 10Gb/s or lower-speed systems. In a direct detection receiver, its photo-detector only responds to changes in the receiving signal optical power, and cannot extract any phase or frequency information from the optical carrier.

Advantages of Coherent receivers:

(1) Greatly improved receiver sensitivity.

(2) Can extract amplitude, frequency, and phase information from an optical carrier, and consequently can achieve much higher capacity in the same bandwidth.

(3) Its DSP can compensate very large chromatic and polarization mode dispersion due to optical fibers, and eliminate the need for optical dispersion compensators and the associated optical amplifiers.

(4) When using balanced detectors with a high common mode noise rejection ratio (CMRR), not only signal-to-noise ratio (SNR) can be improved further, but also agile wavelength selection can be achieved by LO tuning without the use of an optical filter or demultiplexer.

Q3) Let X(t) be a white Gaussian noise process that is input to an LTI system with transfer function

|H(f)|={21<|f|<20 otherwise

If Y(t) is the output, find P (Y (1) <N0).

A3)

Since X(t)X(t) is a zero-mean Gaussian process, Y(t) is also a zero-mean Gaussian process. SY(f) is given by

Therefore,

= 4.

Thus,

To find P(Y(1)<

= Φ

Q4) Find Error Probability of optimum receivers.

A4) The impulse response of this filter should be choosen as

to yield the optimal formation. The result of the convolution is then

Since f(t) =0 for the output at t=T is then

Which is the required match filter value. Note also that subsequent samples also gives the required integral for later bit intervals. For t=2T

When the input signal x(t) is identically the signal f(t), the output at t=T is

The energy of the signal. If the input signal is Gaussian white noise with PSD , then the PSD at the filter output is

The total output power over all frequencies is therefore

The output of the match filter will be

Here E is energy in f(t) or

And is the component due to noise. This quantity is a zero mean random variable and has a mean square power (variance) equal to

where is the noise power spectral density.

Again defining y=y(T) we see that the analysis for the case of simple (sampled) detection continuous to apply, except that

The value A must now be replaced by E, and

The value must be replaced by

The probability of error for matched filter detection is therefore

The polar binary case proceeds in the same way, except that the output of the matched filter is

y(T)=E+ no(T)

once again we can substitute 2E for A and for into the results for the simple case, so

Q5) Explain random process with suitable explanation.

A5) The random process X(t) is called a white noise process if

SX(f)=N0/2, for all f.

Before going any further, let's calculate the expected power in X(t). We have

E[X(t)2]=∫SX(f)df=∫ N0/2df=∞

Thus, white noise, as defined above, has infinite power! In reality, white noise is in fact an approximation to the noise that is observed in real systems. To better understand the idea, consider the PSDs shown in Figure

white-noise

Fig.:- Part (a): PSD of thermal noise; Part (b) PSD of white noise.

Part (a) in the figure shows what the real PSD of a thermal noise and the PSD of the white noise is shown in Part (b) are approximately the same.

The thermal noise in electronic systems is usually modeled as a white Gaussian noise process. It is usually assumed that it has zero mean μX=0 and is Gaussian.

The random process X(t) is called a white Gaussian noise process if X(t) is a stationary Gaussian random process with zero mean, μX=0, and flat power spectral density,

SX(f)=N0/2, for all f

Since the PSD of a white noise process is given by SX(f)=N0/2, its autocorrelation function is

RX(τ)=F−1{N0/2}=N0/2δ(τ),

where δ(τ) is the dirac delta function

δ(x)= 0, x=0 otherwise 1 for x=1

Q6) Write some advantages of optimum filtering.

A6) Optimum filtering is to acquire the best linear estimate of a desired signal from a measurement. The main issues in optimal filtering contain

• filtering that deals with recovering a desired signal d(n) from a noisy signal (or measurement) x(n);

• prediction that is concerned with predicting a signal d(n+m) for m>0 from observation x(n);

• smoothing that is an a posteriori form of estimation, i.e., estimating d(n+m) for m

Q7) Write few advantages of Coherent receivers.

A7)

(1) Greatly improved receiver sensitivity.

(2) Can extract amplitude, frequency, and phase information from an optical carrier, and consequently can achieve much higher capacity in the same bandwidth.

Q8) Define Basis Vectors.

A8) The set of basis vectors {e1, e2, …,en} of a space are chosen such that: Should be complete or span the vector space: any vector a can be expressed as a linear combination of these vectors.

Each basis vector should be orthogonal to all others

·Each basis vector should be normalized:

· A set of basis vectors satisfying these properties is also said to be a complete orthonormal basis

·In an n-dim space, we can have at most n basis vectors

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