Equivalent Baseband Channel Model

Equivalent Baseband Channel Model#

Equivalent Baseband Model#

The equivalent baseband model simplifies the analysis of a passband communication system by shifting all operations and components to baseband (centered around 0 Hz).

This is achieved by mathematically translating the passband signals and channels to their baseband equivalents, and obtain euqivalent baseband observation/received signa:

\[ \boxed{ y_b(t) = h_b(t) * x_b(t) } \]

where

  • \( y_b(t) \): Received baseband signal.

  • \( h_b(t) \): Equivalent baseband channel response (complex-valued).

  • \( x_b(t) \): Transmitted baseband signal (complex-valued).

  • \(*\): Convolution operation.

Equivalent Baseband Channel \( h_b(t) \)#

To transition from the passband channel \( h(t) \) to its baseband equivalent \( h_b(t) \), we perform frequency shifting and filtering operations.

Time-Domain Relationship#

\[ \boxed{ h_b(t) = \sqrt{2} \cdot h(t) e^{-j2\pi f_c t} \ast h_{LP}(t) } \]

where

  • \( h(t) \): Original passband channel impulse response (real-valued).

  • \( f_c \): Carrier frequency.

  • \( e^{-j2\pi f_c t} \): Complex exponential used for frequency shifting (down-conversion).

  • \( h_{LP}(t) \): Ideal low-pass filter with bandwidth \( \left[-\frac{W}{2}, \frac{W}{2}\right] \).

  • \(\ast\): Convolution operation.

Detail Operations

  1. Frequency Shifting: Multiplying \( h(t) \) by \( e^{-j2\pi f_c t} \) shifts the frequency spectrum of the channel down by the carrier frequency \( f_c \), effectively bringing it to baseband.

  2. Complex Representation: Unlike taking only the real part, retaining the full complex expression ensures that both the in-phase (I) and quadrature (Q) components of the channel are captured.

    This is essential for accurately modeling the channel’s impact on the transmitted signal.

  3. Low-Pass Filtering: Applies an ideal low-pass filter to isolate the baseband components, removing any high-frequency artifacts introduced during frequency shifting.

  4. Scaling Factor \( \sqrt{2} \): Compensates for the energy distribution when dealing with complex signals, ensuring power conservation.

Frequency-Domain Relationship#

\[ \boxed{ H_b(f) = \sqrt{2} \cdot H(f + f_c) \cdot \text{rect}\left( \frac{f}{W} \right) } \]

where

  • \( H(f) \): Fourier Transform of the passband channel \( h(t) \).

  • \( H_b(f) \): Fourier Transform of the equivalent baseband channel \( h_b(t) \).

  • \( \text{rect}\left( \frac{f}{W} \right) \): Rectangular function defining the passband of the low-pass filter \( h_{LP}(t) \).

Detail Operations

  1. Frequency Shifting: \( H(f + f_c) \) shifts the spectrum of \( H(f) \) down by \( f_c \), centering it around 0 Hz.

  2. Rectangular Filtering: The \( \text{rect}\left( \frac{f}{W} \right) \) function ensures that only frequencies within \( \left[-\frac{W}{2}, \frac{W}{2}\right] \) pass through, effectively applying the low-pass filter.

  3. Scaling Factor \( \sqrt{2} \): Maintains the correct amplitude scaling after modulation and filtering.