32-level FSK is used with fc = 5000 kHz and fd = 25kHz. Excess bandwidth factor  r= 0.55 . What is the bandwidth efficiency in bps/Hz?

In a Frequency Shift Keying (FSK) system, the bandwidth efficiency (measured in bps/Hz) is a key metric that indicates how effectively the system uses its available bandwidth.

To calculate the bandwidth efficiency, we need to consider the following steps:

### 1. **Determine the Data Rate:**

The number of bits transmitted per symbol is calculated as:

\[
n = \log_2(M)
\]

where \( M = 32 \) (the number of levels in the M-ary FSK system).

So,

\[
n = \log_2(32) = 5 \text{ bits per symbol}
\]

The symbol rate (baud rate) can then be written as:

\[
R_s = \frac{R_b}{n}
\]

where \( R_b \) is the bit rate and \( R_s \) is the symbol rate. Since we are interested in bandwidth efficiency, we focus on how much bandwidth is required to transmit the symbols.

### 2. **Calculate the Bandwidth Required:**

The bandwidth of an M-ary FSK system is calculated using the following formula, considering the excess bandwidth factor \( r \):

\[
B = (1 + r) \times M \times f_d
\]

where:
- \( B \) is the bandwidth.
- \( r = 0.55 \) (excess bandwidth factor).
- \( M = 32 \) (number of levels).
- \( f_d = 25 \text{ kHz} \) (frequency deviation).

Now, substituting the values:

\[
B = (1 + 0.55) \times 32 \times 25 \text{ kHz}
\]
\[
B = 1.55 \times 32 \times 25 \text{ kHz}
\]
\[
B = 1.55 \times 800 \text{ kHz}
\]
\[
B = 1240 \text{ kHz} = 1.24 \text{ MHz}
\]

### 3. **Calculate the Bandwidth Efficiency:**

The bandwidth efficiency \( \eta \) (in bps/Hz) is given by:

\[
\eta = \frac{R_b}{B}
\]

Since each symbol represents 5 bits and the system uses \( M = 32 \) symbols, we assume the bit rate \( R_b \) is proportional to the symbol rate. The number of bits per second is:

\[
R_b = 5 \times f_d = 5 \times 25 \text{ kHz} = 125 \text{ kbps}
\]

Finally, the bandwidth efficiency:

\[
\eta = \frac{R_b}{B} = \frac{125 \text{ kbps}}{1240 \text{ kHz}} \approx 0.1 \text{ bps/Hz}
\]

### Final Answer:

The bandwidth efficiency is approximately **0.1 bps/Hz**.

---

Would you like more details or have any questions about the calculation? 

Here are 5 related questions you might find helpful:
1. How does increasing the number of levels in FSK affect bandwidth efficiency?
2. What is the role of the excess bandwidth factor in FSK systems?
3. Can bandwidth efficiency improve without increasing the data rate?
4. How does FSK compare to PSK or QAM in terms of bandwidth efficiency?
5. What factors influence the choice of \( f_c \) and \( f_d \) in an FSK system?

**Tip:** Increasing the modulation order (e.g., moving from 32-FSK to 64-FSK) can improve data rates but requires more bandwidth. Always balance the trade-offs between bandwidth efficiency and system complexity.

32-level FSK is used with fc = 5000 kHz and fd = 25kHz. Excess bandwidth factor r= 0.55. What is the bandwidth efficiency in bps/Hz?

Calculate the bandwidth efficiency of a 32-level FSK system with fc = 5000 kHz, fd = 25 kHz, and excess bandwidth factor r = 0.55. Learn the detailed step-by-step process for determining the efficiency in bps/Hz, including key concepts in digital communications.

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