# Coherence Bandwidth

**Coherence bandwidth is a concept in wireless communication and signal processing that relates to the frequency range over which a wireless channel remains approximately constant in terms of its characteristics.**

coherence bandwidth is The inverse of Doppler spread delay time, or any spread delay time due to fading in general. The coherence bandwidth is related to the delay spread of the channel, which is a measure of the time it takes for signals to traverse the channel. The two are related by the following formulae:

**Coherence bandwidth = 1/(2*pi*delay spread time)**

**Or, Coherence Bandwidth = 1/(2*pi*root-mean-square delay spread time)**

*(Coherence bandwidth in Hertz)*

*For instance, the coherence bandwidth is 2 MHz when the delay spread is {1/(2*pi*2*10^6)} = 80 ns in a household indoor environment.*

**For narrowband approximation,**

**Coherence Bandwidth = 1/root-mean-square delay spread time**

Coherence bandwidth is a measure of the frequency spread over which a wireless communication channel behaves approximately like a flat fading channel. In other words, it's the bandwidth over which the channel's frequency response remains relatively constant. Coherence bandwidth is a crucial parameter in the design of wireless communication systems, particularly for systems that employ frequency-selective fading channel models.

## Frequency Auto-correlation Function

The frequency auto-correlation function RH(Î”f) of the channel transfer function H(f) is defined as:

**RH(Î”f) = E{H(f)H∗(f+Î”f)}**

**E{⋅}** denotes the expectation operator, and **H*(t)** is the complex conjugate of **H(t)**.

## Coherence Bandwidth Definition

The coherence bandwidth is often defined as the frequency separation Î”f over which the auto-correlation function RH(Î”f) drops to a certain fraction of its maximum value, typically **0.5 or 1/e**:

**Bc≈1/Tm**

Where **Tm** is the multi-path delay spread time or root-mean-square delay spread time, which characterizes the extent of multi-path propagation.

# Coherence Time

## Time Auto-correlation Function

The time auto-correlation function RH(Î”t) of the channel impulse response h(t) is defined as: **RH(Î”t)=E{h(t)h∗(t+Î”t)}**

**E{⋅}**denotes the expectation operator, and

**h∗(t)**is the complex conjugate of

**h(t)**.

## Coherence Time Definition

The coherence time is often defined as the time lag Î”t over which the auto-correlation function RH(Î”t) drops to a certain fraction of its maximum value, typically**0.5 or 1/e**:

**Tc≈1/(2*pi*fD)**

Where

**fD**is the Doppler spread, which characterizes the rate of change of the channel due to relative motion.

**fD**= v*f/c= (30 * 2 * 10^9) / (3 * 10^8) = 200 Hz

**MATLAB Code to find Coherence Time and Coherence Bandwidth**

## Output

Coherence Bandwidth: 498867.5012 Hz

Coherence Time: 2.0045e-06 seconds

**Fig 1: PSD of the generated frequency components**

## Relationship between Coherence Time and Delay Spread

The coherence time of a wireless channel is related to its delay spread. Delay spread refers to the time difference between the arrival of the first and last significant multipath components of a signal. Coherence time represents the time over which the channel's impulse response remains relatively constant, and it's inversely proportional to the delay spread.

The relationship between coherence time **(Tc)** and delay spread **(Td)** can be approximated using the formula:**Tc≈1/Î²⋅Td**

where **Î²** is a factor depending on the specific characteristics of the wireless environment, typically ranging from **2 to 4.**

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**MATLAB Code to find Relationship between Coherence Time and delay Spread **