Power Spectral Density

Power Spectral Density (PSD) describes how the power of a signal is distributed across frequencies. For digital modulation schemes, the PSD determines bandwidth occupancy, spectral efficiency, and adjacent channel interference. Understanding PSD is critical for designing band-limited communication systems and is frequently tested in GATE.

Core Concept of PSD in Digital Modulation

The Power Spectral Density of a signal S(f) is defined as the Fourier transform of its autocorrelation function. For a random digital signal, the PSD gives the average power per unit bandwidth at each frequency. It is measured in Watts per Hz or dBm per Hz. For bandpass modulated signals, the PSD is centered around the carrier frequency fc.

The shape of the PSD depends on the pulse shaping used. For rectangular NRZ pulses of duration T, the Fourier transform is a sinc function, and the PSD is proportional to sinc squared. For a BPSK signal with rectangular pulses and bit period Tb, the PSD has a main lobe extending from fc - Rb to fc + Rb where Rb = 1/Tb is the bit rate. Nulls appear at integer multiples of Rb from the carrier. This gives a null-to-null bandwidth of 2Rb for BPSK.

For QPSK, since two bits are transmitted per symbol and symbol period Ts = 2Tb, the null-to-null bandwidth is 2Rs = 2*(Rb/2) = Rb. Therefore QPSK has half the bandwidth of BPSK for the same bit rate, making it twice as spectrally efficient. OQPSK has the same PSD as QPSK since only timing, not spectral content, is altered.

Mathematical Expression

For a binary PAM signal with equiprobable bits, rectangular pulses, and bit rate Rb = 1/Tb, the baseband PSD is:

Sx(f) = Tb * sinc^2(f * Tb) = (1/Rb) * sinc^2(f/Rb)

The bandpass PSD for BPSK is:

S_BPSK(f) = (Ac^2 * Tb / 4) * [ sinc^2((f - fc)*Tb) + sinc^2((f + fc)*Tb) ]

For QPSK with symbol period Ts = 2Tb and symbol rate Rs = 1/Ts:

S_QPSK(f) = (Ac^2 * Ts / 4) * [ sinc^2((f - fc)*Ts) + sinc^2((f + fc)*Ts) ]

Since Ts = 2Tb, the QPSK main lobe is half the width of BPSK. For M-PSK, the null-to-null bandwidth is 2Rs = 2Rb/log2(M). As M increases, bandwidth decreases but power requirements increase because constellation points come closer together.

MSK has a PSD with faster sidelobe rolloff than QPSK because MSK uses sinusoidal pulse shaping. Its main lobe is slightly wider than QPSK but the sidelobes fall off at 1/f^4 instead of 1/f^2, making it better for adjacent channel interference rejection.

Practical Understanding

In real systems, pulse shaping filters such as the root raised cosine (RRC) filter are applied to reduce bandwidth beyond the ideal sinc-squared case. The raised cosine filter has a rolloff factor alpha (0 to 1) and confines most energy within bandwidth Rs*(1 + alpha)/2 on each side of the carrier. This trades a slightly increased bandwidth for dramatically lower sidelobe levels and no ISI at sampling instants (Nyquist condition).

Spectrum masks are regulatory limits imposed by standards bodies. Transmitted PSD must stay below the mask to avoid interference with adjacent channels. Proper pulse shaping is how designers ensure compliance with these masks while maximizing data rate within the allocated bandwidth.

Given:
Modulation: QPSK with rectangular pulses
Bit rate Rb = 10 Mbps
Carrier frequency fc = 2.4 GHz

Why this formula applies:
QPSK has 2 bits per symbol, so symbol rate Rs = Rb / 2
Null-to-null bandwidth = 2 * Rs

Formula:
Rs = Rb / log2(M) = Rb / 2 (for QPSK, M = 4)
Null-to-null BW = 2 * Rs
First null at fc +/- Rs

Substitution:
Rs = 10 Mbps / 2 = 5 Msymbols/s
Null-to-null BW = 2 * 5 MHz = 10 MHz
First null frequencies: 2400 - 5 = 2395 MHz and 2400 + 5 = 2405 MHz

Comparison with BPSK at same Rb:
BPSK Rs = 10 Msymbols/s, BW = 2 * 10 = 20 MHz

Final Answer:
QPSK null-to-null bandwidth = 10 MHz (half that of BPSK)
BPSK null-to-null bandwidth = 20 MHz
Spectral efficiency of QPSK = Rb / BW = 10/10 = 1 bit/s/Hz

Exam Tip: For GATE PSD questions, always start by finding the symbol rate Rs = Rb / log2(M). Then null-to-null bandwidth = 2Rs. QPSK is twice as spectrally efficient as BPSK at the same bit rate. Also remember that increasing M in M-PSK narrows bandwidth but worsens BEP performance.

PSD Summary by Modulation Scheme

• PSD shape depends on the pulse shaping used. Rectangular pulses give sinc-squared PSD with high sidelobes.
• Null-to-null bandwidth = 2Rs = 2Rb / log2(M) for M-PSK and M-QAM with rectangular pulses.
• Increasing M reduces bandwidth but worsens BEP at the same Eb/N0.
• MSK has faster sidelobe rolloff (1/f^4) than QPSK (1/f^2) due to continuous phase and sinusoidal pulse shaping.
• Root raised cosine (RRC) filtering achieves Nyquist bandwidth with rolloff factor alpha, eliminating ISI at decision instants.

Quick Revision

• PSD = Fourier transform of autocorrelation function. Describes power distribution across frequencies.
• For rectangular pulses: PSD shape = sinc^2. For raised cosine: compact spectrum with no ISI.
• BPSK null-to-null BW = 2Rb. QPSK null-to-null BW = Rb. QPSK is twice as spectrally efficient.
• General formula: null-to-null BW = 2Rs = 2Rb / log2(M).
• MSK: slightly wider main lobe than QPSK but much lower sidelobes, better for adjacent channel protection.
• Trap: PSD depends on pulse shape AND modulation order. Do not confuse null-to-null BW with 3 dB BW or Nyquist BW.
• Raised cosine filter rolloff alpha: BW = Rs*(1 + alpha). At alpha = 0, minimum Nyquist BW = Rs/2.