Channel Impulse Response (CIR)

• 📘 Overview & Theory
• 📘 How does the channel impulse response affect the signal?
• 🧮 MATLAB Codes
• 📚 Further Reading

• Wireless Signal Processing
• CIR, Doppler Shift & Gaussian Random Variable

 The Channel Impulse Response (CIR) is a concept primarily used in the field of telecommunications and signal processing. It provides information about how a communication channel responds to an impulse signal.

 

What is the Channel Impulse Response (CIR) ?

It describes the behavior of a communication channel in response to an impulse signal. In signal processing,  an impulse signal has zero amplitude at all other times and amplitude ∞ at time 0 for the signal. Using a Dirac Delta function, we can approximate this.

 ...(i)

δ(t) now has a very intriguing characteristic. The answer is 1 when the Fourier Transform of δ(t) is calculated.

As a result, all frequencies are responded to equally by δ(t). This is crucial since we never know which frequencies a system will affect when examining an unidentified one. Since it can test the system for all frequencies, δ(t) becomes the perfect option for determining how a system will react.

Channel Impulse Response (CIR) and Multi-path:

If we send a signal in the typical wireless communication medium, that signal will arrive at the receiver as MPCs or multi-paths [Read more]. They arrive at the recipient at different times. They are linear in nature and are delayed variants of the same signal.

The Doppler effect is detected when either the transmitter or receiver or both are moving. The receiving frequency increases as the MS or mobile station approaches the BS or base station. When MS moves away from the receiver, on the other hand, the frequency of receiving decreases.

Channel Impulse Response Equation:

y(t) = x (t ) * h (t) ...(ii)

 

 Where, '*' denotes convolution in time domain

y(t) = Σ x (t - τ) h (t, τ) ...(iii)

A radio channel's time-variant impulse response, where the channel impulse response or channel gain varies with time, is described as h (t). When a signal is sent from the transmitter, it arrives at the receiver with a time delay of x (t -τ ). They are duplicates of the same signal that arrive at the receiver via numerous reflecting or refractive pathways. They're also linear because they're scalar multiples of one another.

The above equation (ii) represents the convolution of the transmitted signal with the channel impulse response. Equation (ii) can be rewritten as y(t) = (h*x)(t), where '*' denotes convolution.

 

How does the channel impulse response affect the signal?

Real-world wireless communication is often modeled as a Linear Time-Invariant (LTI) system, where it is assumed that the channel gain remains constant during the transmission of each symbol. However, channel estimation is frequently performed to track time-varying channel conditions. In this model, the original message bits or symbols are affected by the wireless channel, which can be represented as the convolution of the transmitted signal with the channel’s impulse response. This impulse response accounts for the different path gains caused by multipath propagation. As a result, the receiver does not receive the original signal directly. These multipath components can interfere constructively or destructively, significantly altering the received signal.

Fig: Original Message Signal

Fig: Channel Impulse Response (due to Multi-path or Rayleigh Fading)

Fig: Received Signal after demodulation at the receiver side, which is affected by both rayleigh fading and AWGN noise

Summary

In a Linear Time-Invariant (LTI) system, the output y(t) is given by the convolution of the input signal x(t) with the system's impulse response h(t):

y(t)=x(t)∗h(t)

 

'*' denotes convolution operation in the time domain

When the input signal is an impulse δ(t), the output of the LTI system is the impulse response h(t). This is because the convolution of an impulse with any function returns that function:

δ(t)∗h(t) = h(t)

However, if the input impulse and the received impulse response are not correlated as expected, several factors could be contributing to this discrepancy

 How to calculate bit error rate (BER) from Channel Impulse  Response

To calculate BER versus SNR from a channel impulse response (CIR), you first need to obtain the CIR, which characterizes the effect of the communication channel. Convert the CIR to the frequency domain using the Fourier Transform to get the Channel Frequency Response (CFR). Then, generate a transmitted signal, convolve it with the CIR, and add white Gaussian noise (AWGN) to simulate the received signal. The Signal-to-Noise Ratio (SNR) is calculated as the ratio of the signal power to the noise power, typically expressed in decibels (dB). Demodulate the received signal and compare it with the original transmitted signal to compute the Bit Error Rate (BER)

Deep Dive:

The channel impulse response is calculated using a simple trick. We begin by sending a pilot signal from the transmitter. The data is then retrieved, and the channel Impulse response is calculated. The pilot signal (or bits) are pre-determined. To receive regular updates on channel Impulse Response, we repeat the method in short intervals. The channel Impulse Response is also affected by the environment, such as indoor, outdoor, industrial, residential, etc.

As previously stated, channel impulse response varies depending on the surroundings. For example, channel impulse responses or generated multi-paths are higher in an indoor environment than in an outdoor environment. On the other hand, while comparing different indoor environments, we find that the industrial indoor environment has a higher number of multipath than any other. Because many reflections and refraction on metallic surfaces of heavy equipment, machinery, and other objects generate MPCs in that environment. Compared to MPCs generated outdoors, MPCs formed indoors are closer in time. MPCs are developed outside because of structures, foliage, and other factors. However, compared to indoors, the distance between the transmitter and receiver is greater. As a result, multipath takes longer to reach the receiver than inside.

We generally see clusters in the channel impulse response at higher frequencies (CIR). When MPCs arrive at the receiver and are near in time, they form a cluster. Similarly, there could be several clusters. Let's say we want to send an impulse signal from the transmitter. The signal then travels 100 multipath to reach the receiver. The first 40 MPCs arrive at the receiver in 50 milliseconds, followed by the next 60 MPCs in a 20-millisecond interval, all arriving within 70 milliseconds. The period of the first cluster is 50 milliseconds, and the time duration of the second cluster is 70 milliseconds. And while the time gap between the two clusters is 20 milliseconds, the total duration of the channel impulse response is (50 + 20 + 70) milliseconds.

 

Also, Read the following: 

1. What is convolution (full convolution)
2. Convolution in LTI Wireless Communication Systems
   
3. Equalizer to reduce Multi-path Effects using MATLAB 
4. Channel Impulse Response in the Typical Wireless Communication
5. MATLAB Code for BER vs SNR from Channel Impulse Response
6. Convolution in LTI Wireless Communication Systems
7. Gaussian Random Variable (RV) and its PDF
8. Doppler Shift
9. Fading - Slow & Fast and Large & Small Scale Fading
10. Equalizer - Fundamentals of Channel Estimation  
11.  Impact of Rayleigh Fading and AWGN on Digital Communication Systems
12. Channel Matrix Gain

 

MATLAB code for channel impulse response estimation using FFT-based channel estimation method

 

 

 

 

(Get the MATLAB Code)