# Complex Baseband Equivalent Models

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## 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

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

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

$$s(t) = s_I(t) + j s_Q(t) \;\;\;———— (2)$$

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.

## 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

$$\tilde{y} = \tilde{s} \ast \tilde{h} + \tilde{n} \;\;\;————–(3)$$

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

$$\left(y_I + jy_Q\right) = \left(s_I + js_Q\right) \ast \left(h_I + jh_Q\right) + \left(n_I + jn_Q\right) \;\;\;———- (4)$$

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