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This module outlines the basics of digital communication through a white-noise channel. Transmission consists of pulse shaping and modulation, while reception consists of demodulation, filtering, and sampling. The combined pulse-shaping and filtering designs which prevent inter-symbol interference (ISI) are identified as those satisfying the Nyquist criterion, which is examined in both the time and frequency domains, and the raised-cosine (combined) pulse is given as a common example. The Cauchy-Schwarz inequality is then used to show the matched filtering maximizes the signal-to-noise ratio. Finally, the square-root raised cosine pulse/filter is given as an example of matched filtering that satisfies the Nyquist criterion.

Transmission consists of

  1. pulse shaping: m ˜ ( t ) = n a [ n ] g ( t - n T ) ,
  2. modulation: s ( t ) = Re { m ˜ ( t ) e j 2 π f c t } .

Reception consists of

  1. demodulation: v ˜ ( t ) = LPF { 2 r ( t ) e - j 2 π f c t } ,
  2. filtering: y ( t ) = v ˜ ( t ) * q ( t ) ,
  3. sampling: y [ m ] = y ( m T ) .
This is a two-part flowchart. The first part is title digital modulation, and begins with the expression a[n], then moves to a box titled g(t). Below the box is a circle containing a hook shape, labeled nT. To the right of the box is an arrow pointing to the right, labeled m-tilde(t). This arrow points at an x-circle. Below the x-circle is a circle with a tilde and an arrow pointing back up at the x-circle, labeled e^j2πf_ct.  The middle of this part of the flowchart is bracketed and labeled analog QAM mod. To the right of the x-circle is an arrow pointing to the right at a box labeled Re. To the right of this is an arrow pointing to the right labeled s(t). This arrow points at a cloud labeled channel, which is in the middle of the two parts of the flowchart. The second part of the flowchart is labeled digital demodulation. Beginning on this side to the right of the cloud is an arrow pointing to the right at an x-circle, labeled r(t). Below the x-circle is a circle with a tilde, labeled 2e^-j2πf_ct, with an arrow pointing back up. This section is bracketed and labeled analog QAM demod. To the right of the x-circle is a box labeled LPF. To the right is an arrow labeled v-tilde(t) that points at a box labeled q(t). To the right of this is a line labeled y(t) that is angled upward and disconnected from a final arrow, although an arrow is drawn to show that this disconnected segment is in motion to connect. Below the disconnected portion is another circle with a hook, labeled t = mT. The final arrow points to the right at a final expression, y[m]. This is a two-part flowchart. The first part is title digital modulation, and begins with the expression a[n], then moves to a box titled g(t). Below the box is a circle containing a hook shape, labeled nT. To the right of the box is an arrow pointing to the right, labeled m-tilde(t). This arrow points at an x-circle. Below the x-circle is a circle with a tilde and an arrow pointing back up at the x-circle, labeled e^j2πf_ct.  The middle of this part of the flowchart is bracketed and labeled analog QAM mod. To the right of the x-circle is an arrow pointing to the right at a box labeled Re. To the right of this is an arrow pointing to the right labeled s(t). This arrow points at a cloud labeled channel, which is in the middle of the two parts of the flowchart. The second part of the flowchart is labeled digital demodulation. Beginning on this side to the right of the cloud is an arrow pointing to the right at an x-circle, labeled r(t). Below the x-circle is a circle with a tilde, labeled 2e^-j2πf_ct, with an arrow pointing back up. This section is bracketed and labeled analog QAM demod. To the right of the x-circle is a box labeled LPF. To the right is an arrow labeled v-tilde(t) that points at a box labeled q(t). To the right of this is a line labeled y(t) that is angled upward and disconnected from a final arrow, although an arrow is drawn to show that this disconnected segment is in motion to connect. Below the disconnected portion is another circle with a hook, labeled t = mT. The final arrow points to the right at a final expression, y[m].

Building on analog QAM mod/demod components, digital mod adds pulse shaping&demod adds filtering/sampling.

Simplifying via the complex-baseband equivalent channel:

This figure is a flowchart. It begins with the 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 to the right at a box labeled h-tilde(t). To the right of this is movement to a plus-circle. Below and pointing up at the plus circle is the expression w-tilde(t). To the right of the plus-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 figure is a flowchart. It begins with the 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 to the right at a box labeled h-tilde(t). To the right of this is movement to a plus-circle. Below and pointing up at the plus circle is the expression w-tilde(t). To the right of the plus-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].

Transmitter pulse shaping is used to convert the symbol sequence { a [ n ] } into the continuous message m ˜ ( t ) :

m ˜ ( t ) = n a [ n ] g ( t - n T ) "baseband message" T = "symbol period"

Thus, m ˜ ( t ) can be seen to be a superposition of scaled and time-shifted copies of the pulse waveform g ( t ) .

Example, if the symbol sequence a [ 0 ] , a [ 1 ] , a [ 2 ] , a [ 3 ] , a [ 4 ] equals [ 1 , 3 , - 1 , 1 , 3 ] , then the square pulse g ( t ) shown below left yields the message m ˜ ( t ) shown below right.

This figure is comprised of five aligned graphs in a column and two separate graphs to their right. The graphs plot t against g(t) in the first, g(t - T) in the second, up to g(t - 4T) in the fifth. Horizontal values are labeled in the first quadrant as T, 2T, 3T, 4T, 5T, and 6T. There is one colored box in each graph, occurring down the chart to be centered at 0, T, 2T and so on until 4T. The first chart on the right shows these boxes displayed on the same graph and thus at a different perspective. The graph is labeled a[n]g(t - nT) for n= 0,..., 4. The second graph on the side is the same thing, without the colors, and is labeled m-tilde(t) = Σ_n a[n]g(t - nT). This figure is comprised of five aligned graphs in a column and two separate graphs to their right. The graphs plot t against g(t) in the first, g(t - T) in the second, up to g(t - 4T) in the fifth. Horizontal values are labeled in the first quadrant as T, 2T, 3T, 4T, 5T, and 6T. There is one colored box in each graph, occurring down the chart to be centered at 0, T, 2T and so on until 4T. The first chart on the right shows these boxes displayed on the same graph and thus at a different perspective. The graph is labeled a[n]g(t - nT) for n= 0,..., 4. The second graph on the side is the same thing, without the colors, and is labeled m-tilde(t) = Σ_n a[n]g(t - nT).

Receiver filtering (via q ( t ) ) has two goals:

  1. noise suppression (i.e., SNR improvement),
  2. inter-symbol interference (ISI) prevention.

Noise suppression was briefly discussed in Preliminaries and will soon be revisited in more detail. Next we describe ISI.

Realize that, in the ideal digital comm system, the n t h output y [ n ] would simply equal the n t h input a [ n ] . But in practice, y [ n ] can be corrupted by interference from the other symbols { a [ m ] } m n , known as “inter-symbol interference,” and noise.

Isi-prevention for the noiseless trivial channel

Consider the idealized system

This figure is a flowchart. To begin is the expression a[n] which points at a box labeled g(t). Below the box is a circle containing a hook, labeled nT. To the right of the box is an arrow labeled m-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 figure is a flowchart. To begin is the expression a[n] which points at a box labeled g(t). Below the box is a circle containing a hook, labeled nT. To the right of the box is an arrow labeled m-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].
y ( t ) = q ( τ ) m ˜ ( t - τ ) d τ for m ˜ ( t ) = n a [ n ] g ( t - n T ) = n a [ n ] q ( τ ) g ( t - n T - τ ) d τ = n a [ n ] p ( t - n T ) for p ( t ) = g ( t ) * q ( t ) .

Thus, the idealized system can be re-drawn as

This figure is a flowchart beginning with movement from a[n] to a box labeled p(t). Below the box is a circle containing a hook, labeled nT. 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 figure is a flowchart beginning with movement from a[n] to a box labeled p(t). Below the box is a circle containing a hook, labeled nT. 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].

where

y [ m ] = y ( m T ) = n a [ n ] p ( m T - n T ) = n a [ n ] p ( m - n ) T .

To make y [ m ] = a [ m ] (i.e., prevent ISI), we need

This figure is a graph with a two-part equation to its right. The graph plots t against p(t). A curve of one large wave centered at the vertical axis and two smaller curves on the outside. The curve intersects with the vertical axis at vertical value 1. The curve then decreases on both sides to intersect with the horizontal axis at -T and T. The curve dips to a trough and then increases to intersect with the horizontal axis again at -2T and 2T. Finally, the curve increases slightly, peaks, and then decreases to intersect again with the horizontal axis at -3T and 3T. The equation to the right defines the expression p(mT) to be equal to 1 if m = 0 and equal to 0 if m does not equal zero. This figure is a graph with a two-part equation to its right. The graph plots t against p(t). A curve of one large wave centered at the vertical axis and two smaller curves on the outside. The curve intersects with the vertical axis at vertical value 1. The curve then decreases on both sides to intersect with the horizontal axis at -T and T. The curve dips to a trough and then increases to intersect with the horizontal axis again at -2T and 2T. Finally, the curve increases slightly, peaks, and then decreases to intersect again with the horizontal axis at -3T and 3T. The equation to the right defines the expression p(mT) to be equal to 1 if m = 0 and equal to 0 if m does not equal zero.

which is known as the “Nyquist Criterion.” This criterion can be simply stated as p ( m T ) = δ [ m ] using

δ [ m ] = { 1 m = 0 "distcrete-time impulse," 0 m 0 or "Kronecker delta." a [ n ] = m = - a [ m ] δ [ n - m ] "sifting property."

Examples of Nyquist, and non-Nyquist, combined-pulses p ( t ) for a [ 0 ] , a [ 1 ] , a [ 2 ] , a [ 3 ] , a [ 4 ] = [ 1 , 3 , - 1 , 1 , 3 ] :

This figure consists of two sets of five graphs, each with two graphs to the right connecting the data. The graphs plot t against p(t), p(t - T) and so on until p(t - 4T). The graphs each show a curve that begins with a complete small wave, a large peak, and an identical complete small wave. These waves are centered at 0, T, 2T, and so on unitl 4T for the first set of five graphs. To the right is the composition of all of the graphs together, so that their different sizes and wavelengths are shown in a graph titled a[n]p(t - nT) for n = 0,..., 4. Below this is a simple connection of the strongest occurrences of the graphs, and the graph is titled y(t) = Σ_n a[n]p(t - nT). The second series of five graphs, and the compositions to the right, are similar in construction except that they are all upside-down v-shapes rather than curves, making a much more rigid composition. This figure consists of two sets of five graphs, each with two graphs to the right connecting the data. The graphs plot t against p(t), p(t - T) and so on until p(t - 4T). The graphs each show a curve that begins with a complete small wave, a large peak, and an identical complete small wave. These waves are centered at 0, T, 2T, and so on unitl 4T for the first set of five graphs. To the right is the composition of all of the graphs together, so that their different sizes and wavelengths are shown in a graph titled a[n]p(t - nT) for n = 0,..., 4. Below this is a simple connection of the strongest occurrences of the graphs, and the graph is titled y(t) = Σ_n a[n]p(t - nT). The second series of five graphs, and the compositions to the right, are similar in construction except that they are all upside-down v-shapes rather than curves, making a much more rigid composition.

There is an interesting frequency-domain interpretation. Since

1 T k = - δ f - k T F m = - δ ( t - m T ) ,

we can see that

P ( f ) * 1 T k = - δ f - k T 1 T k = - P f - k T F p ( t ) · m = - δ ( t - m T ) m = - p ( m T ) δ ( t - m T ) .

So, the time-domain Nyquist criterion p ( m T ) = δ [ m ] implies

1 T k = - P f - k T F δ ( t ) ,

which in turn implies

This figure consists of one equation and one graph. The equation reads 1/T Σ_(k = -∞)^∞ P(f - k/T) = 1. The first part of the equation is also the label for the vertical axis of the graph to the right. The horizontal axis is labeled f. The graph is a series of alternating and cyclical peaks and troughs, with each peak and trough occurring at a vertical value of 1 and 0. Two graphs continue simultaneously so that there is a peak and a trough occurring at the same time. These graphs intersect at -3/2T, -1/2T, 1/2T, and 3/2T. This figure consists of one equation and one graph. The equation reads 1/T Σ_(k = -∞)^∞ P(f - k/T) = 1. The first part of the equation is also the label for the vertical axis of the graph to the right. The horizontal axis is labeled f. The graph is a series of alternating and cyclical peaks and troughs, with each peak and trough occurring at a vertical value of 1 and 0. Two graphs continue simultaneously so that there is a peak and a trough occurring at the same time. These graphs intersect at -3/2T, -1/2T, 1/2T, and 3/2T.

In other words, the superposition of 1 T P ( f - k T ) k Z must sum to one. This frequency-domain version of the Nyquist Criterionwill soon come in handy...

A popular choice of combined pulse p ( t ) = g ( t ) * q ( t ) is the “raised-cosine pulse” with rolloff parameter α [ 0 , 1 ] :

p RC ( t ) = cos ( π α t / T ) 1 - ( 2 α t / T ) 2 sinc ( t / T ) , sinc ( x ) : = sin ( π x ) π x P RC ( f ) = T | f | ( 1 - α ) 2 T T cos 2 π T 2 α | f | - 1 - α 2 T ( 1 - α ) 2 T | f | ( 1 + α ) 2 T 0 ( 1 + α ) 2 T | f | .

Tradeoff: larger α less time-spread but more freq-spread:

This figure consists of two graphs, each containing three curves, labeled as α=0, α=0.5, α=1. The first graph shows that these three curves closely follow a wave-like shape with two small waves on the outside and one large peak in the middle at horizontal value 0 and vertical value 1. The graph's horizontal axis is labeled time (T=1). The smaller the alpha-value, the larger the amplitude of the smaller waves will be. The second graph has the same labeled curves except that this graph plots frequency, and the curves follow one large increase, one peak, then one large decrease across the page. The alpha value of 1 is very wavelike, while the alpha of 0.5 is wavelike except that its amplitude is cut off by a horizontal peak that occurs at the maximum value of the graph, 1. The alpha of 0 is a rectangle from -0.5 on the left to 0.5 on the right with height one. This figure consists of two graphs, each containing three curves, labeled as α=0, α=0.5, α=1. The first graph shows that these three curves closely follow a wave-like shape with two small waves on the outside and one large peak in the middle at horizontal value 0 and vertical value 1. The graph's horizontal axis is labeled time (T=1). The smaller the alpha-value, the larger the amplitude of the smaller waves will be. The second graph has the same labeled curves except that this graph plots frequency, and the curves follow one large increase, one peak, then one large decrease across the page. The alpha value of 1 is very wavelike, while the alpha of 0.5 is wavelike except that its amplitude is cut off by a horizontal peak that occurs at the maximum value of the graph, 1. The alpha of 0 is a rectangle from -0.5 on the left to 0.5 on the right with height one.

So, we now know how to design the combined pulse p ( t ) .

But what about the individual pulses g ( t ) and q ( t ) ?

Maximizing snr for isi-free pulses in white noise

Now let's bring the noise back into consideration. Given

This figure is a flowchart. To begin is the expression a[n] which points at a box labeled g(t). To the right of the box is an arrow that points at a circle containing a plus sign. Above the circle is an expression, w-tilde(t), that points down at the plus-circle. To the right of the plus-circle is 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 the label t = mT. 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 figure is a flowchart. To begin is the expression a[n] which points at a box labeled g(t). To the right of the box is an arrow that points at a circle containing a plus sign. Above the circle is an expression, w-tilde(t), that points down at the plus-circle. To the right of the plus-circle is 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 the label t = mT. 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].
E { w ˜ ( t ) w ˜ * ( t - τ ) } = N 0 δ ( τ ) (complex white noise)

we want a { g ( t ) , q ( t ) } pair that maximizes the SNR of y [ m ] .

Separating the noise and signal contributions to y [ m ] via

This is a flowchart of two rows that connect to a final shape and expression. The top row shows movement from w-tile(t) to a box labeled q(t) to a line angled upward that is disconnected from the next arrow and general flow of the chart in the figure. This line is labeled y_n(t). Below the disconnected portion is the label t = mT. There is indication with an angled arrow that the disconnected segment may be moving to connect to the line. The final arrow in this row is labeled y_n[m]. The lower row shows movement from a[n] to a box labeled g(t) to a box labeled q(t). The two boxes are bracketed and labeled p(t). To the right of the boxes is a line angled upward that is disconnected from the next arrow and general flow of the chart in the figure. This line is labeled y_s(t). Below the disconnected portion is the label t = mT. There is indication with an angled arrow that the disconnected segment may be moving to connect to the line. The final arrow in this row is labeled y_s[m]. This is a flowchart of two rows that connect to a final shape and expression. The top row shows movement from w-tile(t) to a box labeled q(t) to a line angled upward that is disconnected from the next arrow and general flow of the chart in the figure. This line is labeled y_n(t). Below the disconnected portion is the label t = mT. There is indication with an angled arrow that the disconnected segment may be moving to connect to the line. The final arrow in this row is labeled y_n[m]. The lower row shows movement from a[n] to a box labeled g(t) to a box labeled q(t). The two boxes are bracketed and labeled p(t). To the right of the boxes is a line angled upward that is disconnected from the next arrow and general flow of the chart in the figure. This line is labeled y_s(t). Below the disconnected portion is the label t = mT. There is indication with an angled arrow that the disconnected segment may be moving to connect to the line. The final arrow in this row is labeled y_s[m].

the SNR can be written

SNR = E s E n = E { | y s [ m ] | 2 } E { | y n [ m ] | 2 } ,

where E s and E n are average signal and noise energies.

Here, we treat both w ˜ ( t ) and a [ n ] as random, implying that y s [ m ] and y n [ m ] are both random.

Notice that, with an ISI-free combined pulse p ( t ) , we get

y s [ m ] = n a [ n ] p ( m - n ) T = a [ m ] p ( 0 ) p ( 0 ) = - q ( τ ) g ( 0 - τ ) d τ ,

so that

E s = E { | y s [ m ] | 2 } = E { a [ m ] - q ( τ ) g ( - τ ) d τ 2 } = E | a [ m ] | 2 σ a 2 - q ( τ ) g ( - τ ) d τ 2 ,

where σ a 2 denotes average symbol energy. Next, notice that

y n [ m ] = y n ( m T ) = - q ( τ ) w ˜ ( m T - τ ) d τ ,

so that

E n = E { | y n [ m ] | 2 } = E - q ( τ ) w ˜ ( m T - τ ) d τ 2 = E - q ( τ ) w ˜ ( m T - τ ) d τ - q * ( τ ' ) w ˜ * ( m T - τ ' ) d τ ' = - q ( τ ) - q * ( τ ' ) E w ˜ ( m T - τ ) w ˜ * ( m T - τ ' ) N 0 δ ( τ ' - τ ) d τ ' d τ = N 0 - | q ( τ ) | 2 d τ .

Putting these together, we find

SNR = σ a 2 N 0 - q ( τ ) g ( - τ ) d τ 2 - | q ( τ ) | 2 d τ .

Cauchy-Schwarz says

- b ( τ ) c ( τ ) d τ 2 - | b ( τ ) | 2 d τ · - | c ( τ ) | 2 d τ with equality iff b ( τ ) = K c * ( τ ) for any K ,

which implies

SNR σ a 2 N 0 - | g ( - τ ) | 2 d τ with equality iff q ( τ ) = K g * ( - τ ) for any K .

Noting that SNR doesn't depend on K , we choose K = 1 . Thus, given pulse g ( t ) , the SNR-maximizing receiver filter is

q ( τ ) = g * ( - τ ) known as a "matched filter".

We can write this in the frequency domain as

Q ( f ) = - q ( τ ) e - j 2 π f τ d τ = - g * ( - τ ) e - j 2 π f τ d τ = - g * ( t ) e j 2 π f t d t = - g ( t ) e - j 2 π f t d t * = G * ( f ) .

Summary: For SNR-maximizing ISI-free pulses, we need

  1. G ( f ) Q ( f ) = P ( f ) satisfying the Nyquist criterion,
  2. Q ( f ) = G * ( f ) ,

which together imply | G ( f ) | 2 must satisfy the Nyquist criterion.

One option is G ( f ) = P RC ( f ) , since P RC ( f ) was Nyquist. We call this the “square-root raised cosine” (SRRC) pulse.Working out the details of F - 1 { G SRRC ( f ) } , we find

g SRRC ( t ) = ( 1 - α ) sinc ( t T ( 1 - α ) ) 1 - ( 4 α t T ) 2 + 4 α cos ( π t T ( 1 + α ) ) π ( 1 - ( 4 α t T ) 2 ) .
This figure consists of two graphs, each containing three curves, labeled as α=0, α=0.5, α=1. The first graph shows that these three curves closely follow a wave-like shape with two small waves on the outside and one large peak in the middle at horizontal value 0 and vertical value 1. The graph's horizontal axis is labeled time (T=1). The smaller the alpha-value, the larger the amplitude of the smaller waves will be. The second graph has the same labeled curves except that this graph plots frequency, and the curves follow one large increase, one peak, then one large decrease across the page. The alpha value of 1 is very wavelike, while the alpha of 0.5 is wavelike except that its amplitude is cut off by a horizontal peak that occurs at the maximum value of the graph, 1. The alpha of 0 is a rectangle from -0.5 on the left to 0.5 on the right with height one. This figure consists of two graphs, each containing three curves, labeled as α=0, α=0.5, α=1. The first graph shows that these three curves closely follow a wave-like shape with two small waves on the outside and one large peak in the middle at horizontal value 0 and vertical value 1. The graph's horizontal axis is labeled time (T=1). The smaller the alpha-value, the larger the amplitude of the smaller waves will be. The second graph has the same labeled curves except that this graph plots frequency, and the curves follow one large increase, one peak, then one large decrease across the page. The alpha value of 1 is very wavelike, while the alpha of 0.5 is wavelike except that its amplitude is cut off by a horizontal peak that occurs at the maximum value of the graph, 1. The alpha of 0 is a rectangle from -0.5 on the left to 0.5 on the right with height one.

At the receiver, we would use q SRRC ( t ) = g SRRC * ( - t ) = g SRRC ( t ) ; the latter equality is due to g SRRC ( t ) being real and symmetric.

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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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