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In this module, we derive the discrete-time implementation of a (baseband-equivalent) digital communication system, which is useful for Matlab simulation and error-rate analysis. To do this, we use the fundamental DSP tool known as "sinc reconstruction", which says that a bandlimited waveform can be represented exactly by a sum of shifted sinc pulses. Sinc reconstruction is separately applied to the transmission pulse, the reception pulse, and the baseband equivalent channel impulse response, after which those three steps can be represented by discrete-time filtering operations. For this, we show that it is essential to sample the filters at least twice as fast as the symbol rate.

Digital implementation of transmitter pulse-shaping and receiver filtering is much more practical than analog. First, recall the analog implementation of our digital communication system: from the module Digital Communication .

This is a flowchart beginning with movement from a[n] to a box labeled g(t). Below the box is a circle with a hook, labeled nT. to the right of the box is an arrow labeled m-tilde(t) that points at box h-tilde(t). To the right of this is an arrow that points at a plus-circle. Below the circle is an expression, w-tilde(t), that points tat the plus-circle. To the right of the circle is an arrow labeled v-tilde(t) that points at a box labeled q(t). To the right of this is a line angled upward that is disconnected from the final arrow and general flow of the chart in the figure. This line is labeled y(t). Below the disconnected portion is a circle with a hook, labeled t = mT, with an arrow pointing back up at the line. There is indication with an angled arrow that the disconnected segment may be moving to connect to the line. The final arrow points at the final expression, y[m]. This is a flowchart beginning with movement from a[n] to a box labeled g(t). Below the box is a circle with a hook, labeled nT. to the right of the box is an arrow labeled m-tilde(t) that points at box h-tilde(t). To the right of this is an arrow that points at a plus-circle. Below the circle is an expression, w-tilde(t), that points tat the plus-circle. To the right of the circle is an arrow labeled v-tilde(t) that points at a box labeled q(t). To the right of this is a line angled upward that is disconnected from the final arrow and general flow of the chart in the figure. This line is labeled y(t). Below the disconnected portion is a circle with a hook, labeled t = mT, with an arrow pointing back up at the line. There is indication with an angled arrow that the disconnected segment may be moving to connect to the line. The final arrow points at the final expression, y[m].

To proceed further, we need an important DSP concept called “sinc reconstruction”:

If waveform x ( t ) is bandlimited to 1 2 T s Hz, then
x ( t ) = n = - x [ n ] sinc 1 T s ( t - n T s ) for x [ n ] = x ( n T s ) .

In other words, a bandlimited waveform can be reconstructed from its samples via sinc pulse shaping.

This figure contains an equation that reads x(t)  = Σ_n x[n] sinc(1/T (t - nT)). Below this equation is a graph of several colors, with each colored curve drawn as a series of waves of different amplitudes and wavelengths. The highest amplitudes are for curves that reach a vertical value of 3. Other various positions in which these curves intersect with each other and the axes are marked, but unlabeled. This figure contains an equation that reads x(t)  = Σ_n x[n] sinc(1/T (t - nT)). Below this equation is a graph of several colors, with each colored curve drawn as a series of waves of different amplitudes and wavelengths. The highest amplitudes are for curves that reach a vertical value of 3. Other various positions in which these curves intersect with each other and the axes are marked, but unlabeled.

Discrete-time pulse-shaping

Applying T P -sampling and reconstruction to g ( t ) (where the SRRC pulse bandwidth 1 + α 2 T requires the use of P 2 ),

g ( τ ) = l g [ l ] sinc ( P T ( τ - l T P ) ) for g [ l ] = g ( l T P ) m ˜ ( t ) = n a [ n ] g ( t - n T ) = n a [ n ] l g [ l ] sinc ( P T ( t - n T - l T P ) ) = n a [ n ] k g [ k - n P ] sinc ( P T ( t - k T P ) ) via k = n P - l = k n a [ n ] g [ k - n P ] := m ˜ [ k ] sinc ( P T ( t - k T P ) )

The sequence m ˜ [ k ] , a weighted sum of P -shifted pulses g [ k ] , can be generated by P -upsampling a [ n ] (i.e., inserting P - 1 zeros between every pair of samples) and filtering with g [ k ] :

This figure contains two graphs and one flowchart. The first graph plots k against a[k]. To the right is a label, P = 4. The graph plots a series of evenly-spaced dots,mostly along the horizontal axis. Some dots are still aligned vertically with proper spacing, but occur above or below the horizontal axis. The first is on the vertical axis, and is labeled a[0]. The second is the sixth dot and is above the horizontal axis, labeled a[1]. The next is the tenth dot, plotted below the horizontal axis, and labeled a[2]. The third is the fourteenth dot, occurring above the horizontal axis, and labeled a[3]. The next graph plots dots, in a similar fashion, except over a graph marked with horizontal axis k and vertical axis g[k]. The dots in this graph follow a sinusoidal shape, from leftmost horizontal value -2P to rightmost 2P. The flowchart below these graphs shows movement from a[n] to a box containing an up arrow and the variable P. Following this is an arrow labeled a-up-arrow[k], which points at a box labeled g[k]. To the right of this is an arrow labeled m-tilde[k] that points to the right at a box labeled sinc(P/T t). Above this box is the label DAC. Below this box is a circle containing a hook and labeled to the right as kT/P, with an arrow from the shape pointing up at the box. To the right of the box is an arrow pointing at a final expression, m-tilde(t). This figure contains two graphs and one flowchart. The first graph plots k against a[k]. To the right is a label, P = 4. The graph plots a series of evenly-spaced dots,mostly along the horizontal axis. Some dots are still aligned vertically with proper spacing, but occur above or below the horizontal axis. The first is on the vertical axis, and is labeled a[0]. The second is the sixth dot and is above the horizontal axis, labeled a[1]. The next is the tenth dot, plotted below the horizontal axis, and labeled a[2]. The third is the fourteenth dot, occurring above the horizontal axis, and labeled a[3]. The next graph plots dots, in a similar fashion, except over a graph marked with horizontal axis k and vertical axis g[k]. The dots in this graph follow a sinusoidal shape, from leftmost horizontal value -2P to rightmost 2P. The flowchart below these graphs shows movement from a[n] to a box containing an up arrow and the variable P. Following this is an arrow labeled a-up-arrow[k], which points at a box labeled g[k]. To the right of this is an arrow labeled m-tilde[k] that points to the right at a box labeled sinc(P/T t). Above this box is the label DAC. Below this box is a circle containing a hook and labeled to the right as kT/P, with an arrow from the shape pointing up at the box. To the right of the box is an arrow pointing at a final expression, m-tilde(t).
sinc-pulse shaping = digital-to-analog conversion (DAC).

Discrete-time receiver filtering

Applying T P -sampling and reconstruction to bandlimited q ( τ ) :

q ( τ ) = l = - q [ l ] sinc P T ( τ - l T P ) for q [ l ] = q ( l T P )

where again we need P 2 , yields

y [ k ] = y ( k T P ) = - q ( τ ) v ˜ ( k T P - τ ) d τ = - l = - q [ l ] sinc P T ( τ - l T P ) v ˜ k T P - τ d τ = l = - q [ l ] - sinc P T τ ' v ˜ ( k - l ) T P - τ ' d τ ' sinc P T t * v ˜ ( t ) t = ( k - l ) T P = v ˜ [ k - l ] = l = - q [ l ] v ˜ [ k - l ] = q [ k ] * v ˜ [ k ] ,

from which y [ m ] is obtained by keeping only every P t h sample:

The flowchart in this graph shows movement from v-tilde(t) to a box containing the expression sinc(P/T t). Above this box is the label AAF. To the right of the box is a line angled upward that is disconnected from the final arrow and general flow of the chart in the figure. Below the disconnected portion is a circle with a hook, labeled k T/P, with an arrow pointing back up at the line. The flowchart continues after the disconnected portion with an arrow labeled v-tilde[k] pointing to the right at a box labeled q[k]. To the right of this is an arrow labeled y-up-arrow[k] pointing to the right at a box containing a down arrow and the variable P. To the right of this is a final arrow pointing at a final expression, y[m]. To the right of this flowchart is a graph plotting f on the horizontal axis and F[sinc (P/T t)]. The graph contains one large rectangle of height T/P and width from horizontal value -P/2T to P/2T. The flowchart in this graph shows movement from v-tilde(t) to a box containing the expression sinc(P/T t). Above this box is the label AAF. To the right of the box is a line angled upward that is disconnected from the final arrow and general flow of the chart in the figure. Below the disconnected portion is a circle with a hook, labeled k T/P, with an arrow pointing back up at the line. The flowchart continues after the disconnected portion with an arrow labeled v-tilde[k] pointing to the right at a box labeled q[k]. To the right of this is an arrow labeled y-up-arrow[k] pointing to the right at a box containing a down arrow and the variable P. To the right of this is a final arrow pointing at a final expression, y[m]. To the right of this flowchart is a graph plotting f on the horizontal axis and F[sinc (P/T t)]. The graph contains one large rectangle of height T/P and width from horizontal value -P/2T to P/2T.

i.e., downsampling. Here sinc ( P T t ) does anti-alias filtering.

Discrete-time complex-baseband channel

Finally, we derive a discrete-time representation of the channel between m ˜ [ k ] and v ˜ [ k ] :

The flowchart in this graph shows movement from m-tilde(k) to a box containing the expression sinc(P/T t). Above this box is the label DAC. To the right of this is an arrow labeled m-tilde(t) that points at a box containing h-tilde(t). To the right of this is an arrow pointing at a plus-circle, and above the circle is the w-tilde(t) pointing down at the circle. To the right of the circle is an arrow labeled v-tilde(t) that points at a box containing the expression sinc(P/T t). Above this box is the label AAF. To the right of the box is a line angled upward that is disconnected from the final arrow and general flow of the chart in the figure. Below the disconnected portion is a circle with a hook, labeled k T/P, with an arrow pointing back up at the line. The flowchart ends after this disconnected segment with an arrow pointing to the right at the expression v-tilde[k]. The flowchart in this graph shows movement from m-tilde(k) to a box containing the expression sinc(P/T t). Above this box is the label DAC. To the right of this is an arrow labeled m-tilde(t) that points at a box containing h-tilde(t). To the right of this is an arrow pointing at a plus-circle, and above the circle is the w-tilde(t) pointing down at the circle. To the right of the circle is an arrow labeled v-tilde(t) that points at a box containing the expression sinc(P/T t). Above this box is the label AAF. To the right of the box is a line angled upward that is disconnected from the final arrow and general flow of the chart in the figure. Below the disconnected portion is a circle with a hook, labeled k T/P, with an arrow pointing back up at the line. The flowchart ends after this disconnected segment with an arrow pointing to the right at the expression v-tilde[k].

Using w ˜ [ k ] to refer to the noise component of v ˜ [ k ] , it can be seen from the block diagram that

w ˜ [ k ] = - w ˜ ( τ ) sinc ( P T ( k T P - τ ) ) d τ

To model the signal component of v ˜ [ k ] , realize that m ˜ [ k ] is effectively pulse-shaped by sinc ( P T t ) * h ˜ ( t ) * sinc ( P T t ) . But since the frequency response of sinc ( P T t ) has a flat gain of T P over the signal bandwidth, and thus the bandwidth of h ˜ ( t ) ,

sinc ( P T t ) * h ˜ ( t ) * sinc ( P T t ) = T 2 P 2 h ˜ ( t ) .

So, with T P -sampling and reconstruction of T 2 P 2 h ˜ ( t ) , i.e.,

T 2 P 2 h ˜ ( t ) = i = - h [ i ] sinc P T ( t - i T P ) for h [ i ] = T 2 P 2 h ( i T P )

we can write v ˜ [ k ] as

v ˜ [ k ] = w ˜ [ k ] + l m ˜ [ l ] T 2 P 2 h ˜ ( k T P - l T P ) = w ˜ [ k ] + l m ˜ [ l ] i h ˜ [ i ] sinc ( k - l - i ) δ [ k - l - i ] = w ˜ [ k ] + l m ˜ [ l ] h ˜ [ k - l ]

yielding the discrete-time channel

This figure is a small flowchart showing movement from m-tilde[k] to a box containing h-tilde[k] to a plus-circle to v-tilde[k]. Above the plus-circle is the expression w-tilde[k] with an arrow pointing down. This figure is a small flowchart showing movement from m-tilde[k] to a box containing h-tilde[k] to a plus-circle to v-tilde[k]. Above the plus-circle is the expression w-tilde[k] with an arrow pointing down.

Merging the discrete-time channel with the discrete-time modulator and demodulator yields

This flowchart shows movement from a[m] to a box containing an up arrow and the variable P, then with an arrow labeled a-up-arrow[k] shows movement to the right to a box labeled g[k]. To the right of this is an arrow labeled m-tilde[k] that points to the right at a box labeled h-tilde[k]. To the right of this is an arrow that points to the right at a plus-circle. Above the plus-circle is the expression w-tilde[k], which points down at the circle. To the right of the plus-circle is an arrow labeled v-tilde[k] that points to the right at a box labeled q[k]. To the right of this is an arrow labeled y-up-arrow[k] that points to the right at a box containing a down arrow and the variable P. To the right of this is an arrow that points to the right at a final expression, y[m]. This flowchart shows movement from a[m] to a box containing an up arrow and the variable P, then with an arrow labeled a-up-arrow[k] shows movement to the right to a box labeled g[k]. To the right of this is an arrow labeled m-tilde[k] that points to the right at a box labeled h-tilde[k]. To the right of this is an arrow that points to the right at a plus-circle. Above the plus-circle is the expression w-tilde[k], which points down at the circle. To the right of the plus-circle is an arrow labeled v-tilde[k] that points to the right at a box labeled q[k]. To the right of this is an arrow labeled y-up-arrow[k] that points to the right at a box containing a down arrow and the variable P. To the right of this is an arrow that points to the right at a final expression, y[m].

known as the “fractionally sampled” system model. This model is very convenient for MATLAB simulationand acts as a foundation for further analysis.

Questions & Answers

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Source:  OpenStax, Introduction to analog and digital communications. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10968/1.2
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