Complex Baseband Equivalent Models

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Key focus: Complex Baseband Equivalent Models are behavioral models that simplify the simulation, saves computation memory requirements and run time.

This article is part of the book Digital Modulations using Matlab : Build Simulation Models from Scratch

Introduction

The passband model and equivalent baseband model are fundamental models for simulating a communication system. In the passband model, also called as waveform simulation model, the transmitted signal, channel noise and the received signal are all represented by samples of waveforms. Since every detail of the RF carrier gets simulated, it consumes more memory and time. In the case of discrete-time equivalent baseband model, only the value of a symbol at the symbol-sampling time instant is considered. Therefore, it consumes less memory and yields results in a very short span of time when compared to the passband models. Such models operate near zero frequency, suppressing the RF carrier and hence the number of samples required for simulation is greatly reduced. They are more suitable for performance analysis simulations. If the behavior of the system is well understood, the model can be simplified further.

Complex baseband representation of a modulated signal

By definition, a passband signal is a signal whose one-sided energy spectrum is centered on non-zero carrier frequency f_c and does not extend to DC. A passband signal or any digitally modulated RF waveform is represented as

\tilde{s}(t) = a(t) cos \left[ 2 \pi f_c t + \phi(t) \right] = s_I(t) cos(2 \pi f_c t) - s_Q(t) sin(2 \pi f_c t) \quad (1)

where,

a(t) = \sqrt{s_I(t)^2 + s_Q(t)^2} \;\; \text{and} \;\; \phi(t) = tan^{-1} \left( \frac{s_Q(t)}{s_I(t)} \right)

Recognizing that the sine and cosine terms in the equation (1) are orthogonal components with respect to each other, the signal can be represented in complex form as

s(t) = s_I(t) + j s_Q(t) \quad\quad (2)

When represented in this form, the signal s(t) is called the complex envelope or the complex baseband equivalent representation of the real signal \tilde{s}(t). The components s_I(t) and s_Q(t) are called inphase and quadrature components respectively. Comparing equations (1) and (2), it is evident that in the complex baseband equivalent representation, the carrier frequency is suppressed. This greatly reduces both the sampling frequency requirements and the memory needed for simulating the model. Furthermore, equation (1), provides a practical way to convert a passband signal to its baseband equivalent and vice-versa [Proakis], as illustrated in Figure 1.

Conversion from baseband to passband and vice-versa
Figure 1: Conversion from baseband to passband and vice-versa

Complex baseband representation of channel response

In the typical communication system model shown in Figure 2(a), the signals represented are real passband signals. The digitally modulated signal \tilde{s}(t) occupies a band-limited spectrum around the carrier frequency (f_c). The channel \tilde{h}(t) is modeled as a linear time invariant system which is also band-limited in nature. The effect of the channel on the modulated signal is represented as linear convolution (denoted by the \ast operator). Then, the received signal is given by

\tilde{y} = \tilde{s} \ast \tilde{h} + \tilde{n} \quad\quad(3)

The corresponding complex baseband equivalent (Figure 2(b)) is expressed as

\left(y_I + jy_Q\right) = \left(s_I + js_Q\right) \ast \left(h_I + jh_Q\right) + \left(n_I + jn_Q\right) \quad\quad (4)

Passband channel model and its baseband equivalent
Figure 2: Passband channel model and its baseband equivalent

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Reference:

[Proakis] Proakis J.G, Digital Communications, fifth edition, New York, McGraw–Hill, 2008

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Topics in this chapter

Digital Modulators and Demodulators - Passband Simulation Models
● Introduction
● Binary Phase Shift Keying (BPSK)
 □ BPSK transmitter
 □ BPSK receiver
 □ End-to-end simulation
● Coherent detection of Differentially Encoded BPSK (DEBPSK)
● Differential BPSK (D-BPSK)
 □ Sub-optimum receiver for DBPSK
 □ Optimum noncoherent receiver for DBPSK
● Quadrature Phase Shift Keying (QPSK)
 □ QPSK transmitter
 □ QPSK receiver
 □ Performance simulation over AWGN
● Offset QPSK (O-QPSK)
● π/p=4-DQPSK
● Continuous Phase Modulation (CPM)
 □ Motivation behind CPM
 □ Continuous Phase Frequency Shift Keying (CPFSK) modulation
 □ Minimum Shift Keying (MSK)
● Investigating phase transition properties
● Power Spectral Density (PSD) plots
● Gaussian Minimum Shift Keying (GMSK)
 □ Pre-modulation Gaussian Low Pass Filter
 □ Quadrature implementation of GMSK modulator
 □ GMSK spectra
 □ GMSK demodulator
 □ Performance
● Frequency Shift Keying (FSK)
 □ Binary-FSK (BFSK)
 □ Orthogonality condition for non-coherent BFSK detection
 □ Orthogonality condition for coherent BFSK
 □ Modulator
 □ Coherent Demodulator
 □ Non-coherent Demodulator
 □ Performance simulation
 □ Power spectral density

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