Minimum Shift Keying

Why MSK?
 Why MSK?

si (t ) = si11 (t ) + si2 2 (t )
s(t ) = sI (t )cos(2πfct ) - sQ (t )sin(2πfct )

Odd Even
1 0
0 0
0 1
1 1
• Power spectral density of QPSK is

4 𝐸𝑏 𝑆𝑖𝑛𝑐 2 2 𝑇𝑏 𝑓
• Side lobe level is sufficiently large

• The wide spectrum of QPSK is due to the character of baseband signal.

• The baseband signal consists of abrupt changes which gives rise to

spectral components at high frequencies

• If we pass the baseband signal through a low pass filter to suppress the
side lobes, then such filtering will cause inter-symbol interference

• In QPSK phase can change by 180 degree at the symbol rate of 1/Ts

• In OQPSK (Offset QPSK), the phase can change 90 degree at the bit rate
1/Tb
• When such waveforms (QPSK ot OQPSK) with abrupt
phase changes are filtered
– the effect of the filter at the time of the abrupt phase
changes is to cause substantial changes in the amplitude of
the waveform.
• At the output of transmitter, we have non-linear
power amplifier.
– These non-linear amplifier suppress the amplitude
variations at the cost of generating spectral components
outside the main lobe and hence in this way, it undo the
effect of filtering
 Solution
• If we remove these abrupt changes, then we can reduce ICI.

• The technique of “smoothing” the waveform is called “MSK”

• In MSK, the baseband waveforms is much smoother than the

abrupt rectangular waveform of QPSK

• The spectrum of MSK has main center lobe which is 1.5 times as
wide as the main lobe of QPSK.

• The side lobes in MSK is much smaller.

• The waveform of MSK exhibits phase continuity.

 • Power spectral density (PSD) of QPSK is
𝟒 𝑬𝒃 𝑺𝒊𝒏𝒄𝟐 𝟐 𝑻𝒃 𝒇
• Power spectral density of MSK is
𝟐
𝟑𝟐 𝑬𝒃 𝑪𝒐𝒔 𝟐 𝝅 𝑻𝒃 𝒇
𝝅𝟐 𝟏𝟔 𝑻𝟐𝒃 𝒇𝟐 − 𝟏

• PSD of QPSK falls off as the inverse square of the frequency

• PSD of MSK falls off as the inverse fourth power of the

frequency
 Generation
• First Generate Even and Odd Bit (similar to QPSK)
• Each bit in both streams (odd and even) is held for the two bit
intervals (2Tb)
• Changes in odd and even bit stream do not occur at same
time.
• Then generate the waveforms 𝑆𝑖𝑛 2𝜋𝑡/4𝑇𝑏 and 𝐶𝑜𝑠ሺ2𝜋𝑡/
4𝑇𝑏 ሻ
• 𝑆𝑖𝑛 2𝜋𝑡/4𝑇𝑏 passes through zero at the end of the symbol
time in even bit
• 𝐶𝑜𝑠 2𝜋𝑡/4𝑇𝑏 passes through zero at the end of the symbol
time in odd bit
• The time period of these waveforms are 4 times of Tb.
 Generation
• Then multiply even bit with 𝑆𝑖𝑛 2𝜋𝑡/4𝑇𝑏 and odd bit with
𝐶𝑜𝑠 2𝜋𝑡/4𝑇𝑏
• Now our modified baseband signal is 2 𝐸𝑏 𝑏𝑒 𝑡 𝑆𝑖𝑛ሺ2𝜋𝑡/
4𝑇𝑏 ሻ and 2 𝐸𝑏 𝑏𝑜 𝑡 𝐶𝑜𝑠 2𝜋𝑡/4𝑇𝑏

• The MSK signal is now written as

2𝜋𝑡
2 𝐸𝑏 𝑏𝑒 𝑡 𝑆𝑖𝑛 𝐶𝑜𝑠 2𝑝𝑖 𝑓𝑐 𝑡 + 2 𝐸𝑏 𝑏𝑜 𝑡 𝐶𝑜𝑠 2𝜋𝑡/4𝑇𝑏 𝐶𝑜𝑠 2𝑝𝑖 𝑓𝑐 𝑡
4𝑇𝑏
 CPFSK
Consider a continuous-phase frequency-shift keying (CPFSK) signal, which is
defined for the signaling interval

(1)

• where Eb is the transmitted signal energy per bit and Tb is the bit duration.
• This new term, denoting the value of the phase at time t = 0, sums up the
past history of the FM process up to time t = 0. The frequencies f1 and f2
are sent in response to binary symbols 1 and 0, respectively, applied to the
modulator input.
 Cont’d
Representing the CPFSK signal s(t) is to express it as a conventional angle-
modulated signal

(2)

• where θ(t) is the phase of s(t) at time t. When the phase θ(t) is a continuous
function of time, the modulated signal s(t) is itself also continuous at all times,
including the inter-bit switching times.
• The phase θ(t) of a CPFSK signal increases or decreases linearly with time
during each bit duration of Tb seconds, as shown by

(3)

• where the plus sign corresponds to sending symbol 1 and the minus sign
corresponds to sending symbol 0; the dimensionless parameter h is to be
defined.
 2

Substituting (3) into (2)

2𝐸𝑏 ℎ
𝑠 𝑡 = 𝐶𝑜𝑠 2𝜋𝑡 𝑓𝑐 + 𝜃 0 ±
𝑇𝑏 2𝑇𝑏

The nominal carrier frequency fc is, therefore, the arithmetic mean of the
transmitted frequencies f1 and f2.

The difference between the frequencies f1 and f2, normalized with respect to the
bit rate 1/Tb, defines the dimensionless parameter h, which is referred to as the
deviation ratio.
 Phase Trellis
From (3) we find that, at time t = Tb,

• i.e. sending symbol 1 increases the phase of a CPFSK signal s(t) by πh

radians, whereas sending symbol 0 reduces it by an equal amount.

The variation of phase θ(t) with time t

follows a path consisting of a sequence
of straight lines, the slopes of which
represent frequency changes.

The tree makes clear the transitions of

phase across successive signaling
intervals.
 Phase Trellis
For example, the path shown in boldface corresponds to the binary sequence
1101000 with θ(0) = 0. Henceforth, we focus on h = ½

• With h = 1/2, the frequency deviation (i.e., the difference between the
two signaling frequencies f1 and f2) equals half the bit rate.
• The frequency deviation h = 1/2 is the minimum frequency spacing that
allows the two FSK signals representing symbols 1 and 0 to be coherently
orthogonal.
• It is for this reason that a CPFSK signal with a deviation ratio of one-half is
commonly referred to as minimum shift-keying (MSK)
 Minimum Frequency Deviation

• Although any two distinct frequencies f0 and f1 can be used for

communication purpose, it greatly helps in receiver design if the two
distinct signals are orthogonal to each other, i.e.,

• How close can the two frequencies should be? Or in other words, what is
the smallest possible value of ΔF? As closer the two frequencies, the more
the number of channels available for other users in the same spectrum.
 Minimum Frequency Deviation

• Since , first term goes to zero, thus resulted into

Minimum Frequency Deviation
MSK
Expanding the CPFSK signal s(t) in terms of its in-phase and quadrature
components as
2𝐸𝑏 2𝐸𝑏
𝑠 𝑡 = cos 𝜃 𝑡 cos 2𝜋𝑓𝑐 𝑡 − sin 𝜃 𝑡 sin 2𝜋𝑓𝑐 𝑡
𝑇𝑏 𝑇𝑏
With the deviation ratio h = 1/2, from (3),

𝜋
𝜃 𝑡 =𝜃 0 ± 𝑡, 0 ≤ 𝑡 ≤ 𝑇𝑏
2 𝑇𝑏

• where the plus sign corresponds to symbol-1 and the minus sign
corresponds to symbol-0.
• Since the phase θ(0) is 0 or π depending on the past history of the
modulation process, we find that in the interval –Tb ≤ t ≤ Tb, the polarity of
cosθ(t) depends only on θ(0), regardless of the sequence of 1s and 0s
transmitted before or after t = 0.
 Signal-Space Diagram of MSK
2𝐸𝑏 2𝐸𝑏
𝑠 𝑡 = cos 𝜃 𝑡 cos 2𝜋𝑓𝑐 𝑡 − sin 𝜃 𝑡 sin 2𝜋𝑓𝑐 𝑡
𝑇𝑏 𝑇𝑏
𝑠 𝑡 = 𝑠𝐼 𝑡 cos 2𝜋𝑓𝑐 𝑡 − 𝑠𝑄 𝑡 sin 2𝜋𝑓𝑐 𝑡
Signal-Space Diagram of MSK
 Signal-Space Diagram of MSK
The two orthonormal basis functions φ1(t) and φ2(t) characterizing the generation
of MSK are defined by the following pair of sinusoidally modulated quadrature
carriers:

However
𝑠 𝑡 = 𝑠𝐼 𝑡 cos 2𝜋𝑓𝑐 𝑡 − 𝑠𝑄 𝑡 sin 2𝜋𝑓𝑐 𝑡

2𝐸𝑏 𝜋𝑡 2𝐸𝑏 𝜋𝑡
𝑠 𝑡 = cos 𝜃 0 cos cos 2𝜋𝑓𝑐 𝑡 − sin 𝜃 𝑇𝑏 𝑠𝑖𝑛 sin 2𝜋𝑓𝑐 𝑡
𝑇𝑏 2𝑇𝑏 𝑇𝑏 2𝑇𝑏

𝑠 𝑡 = 𝑠1 𝜙1 𝑡 + 𝑠2 𝜙2 𝑡
Thus
𝑠1 = 𝐸𝑏 cos 𝜃 0 = ± 𝐸𝑏 𝑠2 = − 𝐸𝑏 sin 𝜃 𝑇𝑏 = ± 𝐸𝑏
 Signal-Space Diagram of MSK

𝑠1 = 𝐸𝑏 cos 𝜃 0 = ± 𝐸𝑏
𝑠2 = − 𝐸𝑏 sin 𝜃 𝑇𝑏 = ± 𝐸𝑏

𝑠 𝑡 = 𝑠1 𝜙1 𝑡 + 𝑠2 𝜙2 𝑡
Generation and Coherent Detection of MSK Signals
Generation and Coherent Detection of MSK Signals
Generation and Coherent Detection of MSK Signals