Doppler Delay (with MATLAB + Simulator)

Doppler Shift Formula

When either the transmitter or the receiver is in motion, or when both are in motion, Doppler Shift is an essential parameter in wireless communication. We notice variations in reception frequencies in vehicles, trains, or other similar environments. In plain language, the received signal frequency increases as the receiver moves toward the transmitter and drops as the receiver moves in the opposite direction. This phenomenon is called the Doppler shift or Doppler spread.

Doppler Shift Formula

The relationship between the transmitted and received frequency is defined as:

fR = fT ± fD

fR = Receiving frequency
fT = Transmitted frequency
fD = Doppler frequency (shift)

The Doppler frequency is determined by the speed of the vehicle and the angle of the arriving wave:

Doppler shift (LOS path): ν1 = fc * v / c
Doppler shift (Reflected path): ν2 = (fc * v / c) * cos(θ)

Doppler spread: BD = | ν2 − ν1 |

Where c is the speed of light and v is the speed of the vehicle. Alternatively, expressed via wavelength (λ):

BD = | (v / λ) * (cos(θ) - 1) |

Movement Scenarios:

1. Towards BS: When the MS reaches towards the Base Station, θ = 0°, so cos(0°) = 1. The frequency increases:
fR = fT + (v / λ)

2. Away from BS: When the MS moves away, θ = 180°, so cos(180°) = -1. The frequency decreases:
fR = fT - (v / λ)

Doppler Shift vs Doppler Spread

1. Doppler Shift

Represents a single frequency change due to relative motion between a source and an observer. For example, the frequency offset of a radar signal reflecting off a single moving car.

2. Doppler Spread

The range of frequency shifts occurring in a multipath environment. It measures the spectral broadening and affects the channel's coherence time.

MATLAB Implementation

% Doppler Shift calculation script
clc; clear; close all;

c = 3e8;      % Speed of light (m/s)
f0 = 2.4e9;   % Center frequency (2.4 GHz)

% Velocities of different objects
velocities = [-30, -10, 0, 10, 20]; 

% Calculate Shifts and Spread
doppler_shifts = (velocities / c) * f0;
doppler_spread = max(doppler_shifts) - min(doppler_shifts);

fprintf('Doppler Shifts (Hz):\n');
disp(doppler_shifts);
fprintf('Doppler Spread (Hz): %f\n', doppler_spread);

Expected Output:

Relative velocities (in m/s): -30 -10 0 10 20 Doppler Shifts (Hz): -240 -80 0 80 160 Doppler Spread (Hz): 400.000000

How Doppler Spread Affects Communication

Doppler spread causes fading in wireless communication. When received power fluctuates over time, it results in fast or slow fading. High Doppler spread leads to a short coherence time, meaning the channel changes very rapidly.

For practical systems, the received signal r(t) is modeled as:

r(t) = s(t - τ) * h(τ, t) * exp(j * 2 * π * fD * (t - τ))

s(t - τ): Transmitted signal with delay τ.
h(τ, t): Time-varying impulse response.
exp(j...): Complex phase shift due to Doppler frequency fD.

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