Signals and Systems

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California State University, Bakersfield

Hani Mehrpouyan1,

Department of Electrical and Computer Engineering,

California State University, Bakersfield

Lecture 12 (Modulations)

February 1st, 2013

1 Some of the lectures notes here reproduced are taken from course textbooks: “Digital Communications: Fundamentals and Applications” B. Sklar. “Communication Systems Engineering”, J. G. Proakis and M Salehi, and “Lecture Notes for Digital Communication, Queen’s University, Canada”, S. Yousefi.

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California State University, Bakersfield

Outline

� Amplitude-Shift Keying (ASK)

� Geometrical Representation of PAM and ASK

� Higher-dimensional Constellations

� Two-dimensional Constellations

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California State University, Bakersfield

Amplitude-Shift Keying (ASK)

� PAM, as discussed, can be both baseband and bandpass. Another name for bandpass PAM with a rectangular pulse shape (only) is Amplitude-Shift Keying (ASK) modulation.

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Geometrical Representation of PAM and ASK

� It is obvious that both M-PAM and M-ASK only require one basis function for their geometrical representation, i.e., are one-dimensional digital modulation schemes.

� Baseband PAM:

� Error probability (as mentioned before) will be related to the Euclidean distance between the points (corresponding to the signals) in the signal constellation. In 1-D cases like PAM and ASK, the (Euclidean) distance between two signals sm(t) and sn(t) is:

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Geometrical Representation of PAM and ASK

� For symmetric 4-PAM:

� Signals are symmetrically located around the origin and are equi-distance.

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Geometrical Representation of PAM and ASK

� Average energy for the equiprobable case:

� Generally for symmetric M-PAM, the M magnitudes are:

� Average energy:

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California State University, Bakersfield

Geometrical Representation of PAM and ASK

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PAM and ASK

Note: all the above also applies to the M-ASK= bandpassM-PAM

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PAM and ASK

� Remark: With the formulation of symmetrical PAM and ASK with Am = (2m − 1 − M), the Minimum Euclidean Distance (MED) between the points of the resulting constellation remain constant (regardless of M):

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Two-dimensional Constellations� Two-dimensional constellations have received a lot of attention in numerous applications. This is due to the fact that they offer attractive trade-offs among BandWidthEfficiency (BWE), Power Efficiency (PE), and, Cost & Complexity.

� Reminder: increasing BWE and PE are conflicting design objectives.

� Use of efficient modulation techniques: traditionally spearheaded by the telecom industry since the traditional telephone companies’ foremost resource consists of sharply band-limited voice-grade channels.

� Typical telephone channel is characterized with a high Signal-to-Noise Ratio (SNR) of 30 dB and a bandwidth of 3 KHz.

� Note: PSK= Phase-Shift Keying, and QAM = Quadrature Amplitude Modulation (2-D schemes).

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2-D Baseband Signals� Reminder: Two signals s1(t) and s2(t) are orthogonal over an interval [0, T] if:

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California State University, Bakersfield

2-D Baseband Signals

� First pair: orthogonal as they carefully alternate resulting in zero correlation: Walsh or Hadamard Signals.

� Second pair: orthogonal as they exist in non-overlapping intervals. This is also a common method to choose orthogonal functions over a period T.

� One needs only to normalize the orthogonal functions in order to form an orthonormal basis.

� Example: Consider the following 2 signals. Find an orthonormal basis for them and from that find the corresponding signal constellation. What is the distance between the two signals in the geometrical sense (Euclidean)?

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2-D Baseband Signals

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2-D Baseband Signals

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California State University, Bakersfield

2-D Baseband Signals

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California State University, Bakersfield

2-D Baseband Signals

Alternately, we can use these basis functions:

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California State University, Bakersfield

2-D Baseband Signals

Remark 1: The distance between the 2 points in both constellations remains the same.

Remark 2: The energy of each point in both cases is the same.

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California State University, Bakersfield

Orthogonal and BiorthogonalSignals/Constellations

� Given a set of orthogonal signals {si(t)}Mi=1, one can construct a biorthogonal signal set by adding the negative of all the signals to the set:

� Result: Dimensionality stays the same while the number of signals doubles. That is: BWE increases by 1 unit (bit per sec per Hz) .

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California State University, Bakersfield

Orthogonal and BiorthogonalSignals/Constellations

� Note: if the underlying orthogonal signal set is equi-energy, so is the resulting biorthogonal signal set.

� Significance of equi-energy constellations: they are ideal for cases where we suffer from much magnitude distortion (fading and shadowing). They result in simpler hardware design and gain control.

� What do we need to do if we would like to increase the BWE even beyond the biorthogonal case?

� We can keep the bandwidth usage and complexity almost fixed by fixing the dimensionality.

� Let us now to explore other configurations in two dimensions.