DSSS-CDMA 

This explanation builds intuition step by step for DSSS-CDMA, chip rate, processing gain, and how multiple users share the same analog carrier.

What DSSS-CDMA Really Is

• All users transmit at the same time
• Over the same frequency band
• Using the same RF carrier
• Separated using codes

FDMA → Different frequencies

TDMA → Different time slots

CDMA → Same time + same frequency, different codes

Role of the Carrier Signal

There is one common carrier frequency for all users.

Carrier frequency: 900 MHz

Bandwidth: 1.25 MHz (IS-95)

The carrier is NOT assigned per user.

s_k(t) = b_k(t) · c_k(t) · cos(2πf_ct)

• b_k(t): user data (low rate)
• c_k(t): spreading code (high rate)
• f_c: same carrier for everyone

Chip and Chip Rate

• Bit → Information (0 or 1)
• Chip → Element of spreading code (+1 / −1)

Bit rate R_b = 10 kbps

Code length N = 100

Chip rate R_c = N × R_b = 1 Mcps

Processing Gain

PG = R_c / R_b = Spread Bandwidth / Data Bandwidth

R_b = 10 kbps

R_c = 1 Mcps

PG = 100 = 20 dB

Numerical DSSS-CDMA Example

User data:

b₁ = [ 1, -1 ]

b₂ = [ -1, 1 ]

    

Spreading codes (N = 4):

c₁ = [ 1, 1, -1, -1 ]

c₂ = [ 1, -1, 1, -1 ]

c₁ · c₂ = 0 (orthogonal)

    

Spread signals:

s₁ = [ 1, 1, -1, -1, -1, -1, 1, 1 ]

s₂ = [ -1, 1, -1, 1, 1, -1, 1, -1 ]

    

Channel sum:

r = [ 0, 2, -2, 0, 0, -2, 2, 0 ]

    

Intuition

Concept	Mathematical Meaning
User separation	Code orthogonality
Same carrier	Same cos(2πfct)
Same bandwidth	Chip rate
Interference rejection	Inner product ≈ 0
Processing gain	Chip length N

DSSS-CDMA Demodulation Process

At the receiver, the base station performs coherent demodulation followed by despreading to recover the desired user’s data.

Received signal:
r(t) = Σ b_k(t)c_k(t)cos(2πf_ct)

Step 1: Carrier Demodulation

r(t) × cos(2πf_ct) → baseband + high-frequency terms

Step 2: Low-Pass Filtering (LPF)

Removes 2f_c components, leaving:

r_bb(t) ≈ Σ b_k(t)c_k(t)

Step 3: Despreading (Code Correlation)

r_bb(t) × c_desired(t)

Desired user adds coherently; other users average to zero.

Step 4: Integrate over One Bit Period

∫ r_bb(t)c_k(t) dt ≈ N · b_k

Step 5: Decision Device

If output > 0 → bit = +1
If output < 0 → bit = −1

This completes carrier demodulation, despreading, and bit recovery.

For Example 

take first 4 symbols of the received signal [0, 2, -2, 0]

then at user 1 perform [0, 2, -2, 0].c1' = [ 1, 1, -1, -1 ].[0, 2, -2, 0]' = 4/4 = 1

at user 2 perform [0, 2, -2, 0].c2' = [0, 2, -2, 0].[ 1, -1, 1, -1 ]' = -4/4 = -1

User 1 spread signal: [ 1  1 -1 -1 -1 -1  1  1 ]

User 2 spread signal: [-1  1 -1  1  1 -1  1 -1 ]

Received (channel sum): [ 0  2 -2  0  0 -2  2  0 ]

Recovered User 1 bits: [ 1 -1 ]

Recovered User 2 bits: [-1  1 ]

Further Reading

1. DSSS-CDMA Simulator