Pulse Shaping

Time Domain and Frequency Domain

• A waveform shows how the signal changes with time
• The Fourier transform gives its frequency-domain description
• Abrupt transitions spread signal energy into sidelobes
  • Rectangular-pulse spectrum (relative power density vs. frequency offset / symbol rate): main lobe between nulls at , sidelobes about 13 dB down, further nulls at

The Same Bits in Time

• Without explicit pulse shaping: abrupt transitions
• Root-raised-cosine (RRC) transmit shaping: smooth, overlapping pulses
• Same eight bits and symbol rate; only the transmit pulse shape changes (normalized baseband voltage, uncoded BPSK, RRC roll-off )
• The pulse overlap is intentional; a matched receiver controls the interference at the decision instants

RF Bandwidth and Sidelobes

• Same symbol rate, 1.0 Msymbol/s:
  • Rectangular: first-null span ; sidelobes extend beyond it
  • Ideal RRC: RF span , with no sidelobes in the plot

RF Bandwidth

• Practical filters need allowances beyond the ideal RRC span

Benefits and Costs

Benefit	Cost or constraint
Lower out-of-band emissions	Transmit and receive filters must work together
More controlled channel bandwidth	Filter memory and timing affect symbol decisions
Closer channel spacing	Conventional RRC-shaped BPSK can need amplifier backoff
More efficient use of spectrum	Choose a waveform compatible with the spacecraft amplifier

• Saturation can distort envelope variations and regrow sidelobes
• GMSK (Gaussian minimum-shift keying) and selected shaped OQPSK (offset QPSK) can offer favorable amplifier behavior (CCSDS 413.0-G-3, Bandwidth-Efficient Modulations)

Bandwidth Example: BPSK and QPSK

Assumptions	BPSK	QPSK
Information rate	1.0 Mbit/s	1.0 Mbit/s
Bits per symbol	1	2
Symbol rate	1.0 Msymbol/s	0.5 Msymbol/s
at	1.35 MHz	0.68 MHz

• Uncoded examples, ; both transmit the same number of information bits each second (symbol rate)

Pulse-Shape Mismatch at the Receiver

• The receive filter should match the transmit shaping; a mismatch costs required : pulse-shape mismatch example
• References: MathWorks, Raised Cosine Filtering and Raised Cosine Receive Filter documentation; Gallager, MIT 6.450 Ch. 6