## Small-Scale Fading

### PROF. MICHAEL TSAI

### 2017/10/30

## Multipath Propagation

**2**

RX just sums up all Multi Path Component (MPC).
## Multipath Channel Impulse Response

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An example of the time-varying discrete-time impulse response for a multipath radio channel

The channel impulse response
when 𝑡 = 𝑡_{#} (what you receive at
the receiver when you send an
impulse at time 𝑡_{#})

𝜏_{#} = 0, and represents the time the first signal arrives at the receiver.

Summed signal of all multipath
components arriving at 𝜏_{'}~𝜏_{')*}.

**Excess delay: the delay with respect **
to the first arriving signal (𝜏)

**Maximum excess delay: the delay **
of latest arriving signal

### Time-Variant Multipath Channel Impulse Response

**4**

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Because the transmitter, the receiver, or the reflectors are moving, the impulse response is time-variant.

**• The channels impulse response is given by:**

**• If assumed time-invariant (over a small-scale time or ** **distance):**

## Multipath Channel Impulse Response

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Phase change due to different arriving time Additional phase change due to reflections

Amplitude change (mainly path loss) Summation over all MPC

**6**

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Following this axis, we study how “spread-out” the impulse response are.

(related to the physical layout of the TX, the RX, and the reflectors at a single time point)

### Two main aspects

### of the wireless

### channel

**7**

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t(t_{3})

### h

_{b}

### (t,t)

Following this axis, we study how “spread-out” the impulse response are.

(related to the physical layout of the TX, the RX, and the reflectors at a single time point)

### Two main aspects

### of the wireless

### channel

## Power delay profile

**• To predict h**

_{b}**(t) a probing pulse p(t) is sent s.t.**

**• Therefore, for small-scale channel modeling, POWER ** **DELAY PROFILE is found by computing the spatial ** **average of |h**

_{B}**(t;t)|**

^{2}**over a local area.**

**8**

### 𝑝 𝑡 ≈ 𝛿(𝑡 − 𝜏)

### 𝑃 𝑡; 𝜏 ≈ 𝑘 ℎ

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### 𝑡; 𝜏

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TX 𝑝(𝑡) RX

Average over

several measurements in a local area

## Example: power delay profile

**9**

From a 900 MHz cellular system in San Francisco

## Example: power delay profile

**10**

Inside a grocery store at 4 GHz

## Time dispersion parameters

**• Power delay profile is a good representation of the ** **average “geometry” of the transmitter, the receiver, ** **and the reflectors.**

**• To quantify “how spread-out” the arriving signals are, ** **we use time dispersion parameters:**

### • Maximum excess delay: the excess delay of the latest arriving MPC

### • Mean excess delay: the “mean” excess delay of all arriving MPC

### • RMS delay spread: the “standard deviation” of the excess delay of all arriving MPC

**11**

Already talked about this

## Time dispersion parameters

**• Mean Excess Delay**

**• RMS Delay Spread**

**12**

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First moment of the power delay profile

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Second moment of the power delay profile Square root of the second central moment of the power delay

profile

## Time dispersion parameters

**• Maximum Excess Delay:**

•

**Original version: the excess delay of the latest arriving MPC**

•

### In practice: the latest arriving could be smaller than the noise

•

### No way to be aware of the “latest”

**• Maximum Excess Delay (practical version):**

•

### The time delay during which multipath energy falls to X dB below the maximum.

**• This X dB threshold could affect the values of the time-** **dispersion parameters**

•

### Used to differentiate the noise and the MPC

•

### Too low: noise is considered to be the MPC

•

### Too high: Some MPC is not detected

**13**

## Example: Time dispersion parameters

**14**

## Coherence Bandwidth

### • **Coherence bandwidth is a statistical measure of the ** **range of frequencies over which the channel can be ** **considered “flat” **

**à a channel passes all spectral components with **

**approximately equal gain and linear phase.**

## Coherence Bandwidth

### • **Bandwidth over which Frequency Correlation ** **function is above 0.9**

### • **Bandwidth over which Frequency Correlation ** **function is above 0.5**

**16**

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Those two are approximations derived from empirical results.

## Typical RMS delay spread values

**17**

## Signal Bandwidth &

## Coherence Bandwidth

**18**

f t

Transmitted Signal

𝑇_{8}

𝑇_{8}: symbol period

𝐵_{8}
𝐵_{8}: signal bandwidth

t_{0}

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_{0}

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_{1}

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_{2}

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_{3}

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_{4}

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_{5}

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### 𝑃(𝑡; 𝜏)

𝑇_{8} ≈ 1
𝐵_{8}

𝐵_{<}

𝜎_{>}

### Frequency-selective fading channel

**19**

𝐵_{<}

𝐵_{8} f

f

### 𝐵

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TX signal

Channel

RX signal

×

= =∗

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𝜎_{>}

𝑇_{8} t

### 𝑇

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### < 𝜎

_{>}

These will become inter- symbol interference!

𝑇_{8}

## Flat fading channel

𝐵_{<}

f
𝐵_{8}

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### 𝐵

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Channel

RX signal

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### 𝑇

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### 𝑃(𝑡; 𝜏)

𝜎_{>}

t
𝑇_{8}

𝑇_{8} No significant ISI

## Equalizer 101

**• An equalizer is usually used in a frequency-selective ** **fading channel**

### • When the coherence bandwidth is low, but we need to use high data rate (high signal bandwidth)

**• Channel is unknown and time-variant**

### • Step 1: TX sends a known signal to the receiver

### • Step 2: the RX uses the TX signal and RX signal to estimate the channel

### • Step 3: TX sends the real data (unknown to the receiver)

### • Step 4: the RX uses the estimated channel to process the RX signal

### • Step 5: once the channel becomes significantly different from the estimated one, return to step 1.

**21**

## Example

0 1 2 3 4 5 -30dB

-20dB -10dB 0dB

t P(t)

Would this channel be suitable for AMPS or GSM without the use of an equalizer?

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## Example

**• Therefore:**

**• Since B**

_{C}**> 30KHz, AMPS would work without an ** **equalizer.**

**• GSM requires 200 KHz BW > B**

_{C}**à An equalizer would be ** **needed.**

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