SCEE08007 Signals and Communications 2

Formula and Tables of Transforms

Trigonometric Identities

sinx =ejx ejx

2jcosx =

ejx + ejx

2

1 = cos2 x+ sin2 x cos 2x = cos2 x sin2 x

cos2 x =1

2(1 + cos 2x) sin2 x =

1

2(1 cos 2x)

cos3 x =1

4(3 cosx+ cos 3x) tan (AB) =

tanA tanB

1 tanA tanB

cos(AB) = cosA cosB sinA sinB sin(AB) = sinA cosB cosA sinB

sinA+ sinB = 2 sin

(A+B

2

)cos

(AB

2

)sinA sinB = 2 cos

(A+B

2

)sin

(AB

2

)

cosA+ cosB = 2 cos

(A+B

2

)cos

(AB

2

)cosA cosB = 2 sin

(A+B

2

)sin

(AB

2

)

cosA cosB =1

2[cos(A+B) + cos(AB)] sinA cosB =

1

2[sin(A+B) + sin(AB)]

sinA sinB =1

2[cos(AB) cos(A+B)]

a cosx+ b sinx = c cos (x+ ) where c =a2 + b2 and = tan1

b

a

g

a

b

c

a

b

Cosine Rule: c2 = a2 + b2 2ab cos

Sine Rule:a

sin=

b

sin=

c

sin

Fourier Series Analysis of Periodic Waveforms

If g(t) is periodic with period T , then:

g(t) =a02

+

n=1

[an cos (n0t) + bn sin (n0t)]

where an =2

T

T2

T

2

g(t) cos (n0t) dt and bn =2

T

T2

T

2

g(t) sin (n0t) dt

or: g(t) =

n=

cn ejn0t where cn =

1

T

T2

T

2

g(t) ejn0t dt

where 0 =2piT

= 2f0; f0 =1

Tis the fundamental frequency.

Fourier Transform Analysis of Aperiodic Signals

The Fourier transform of a signal g(t) is given by:

G() =

g(t) ejt dt and g(t) =1

2

G() ejt d

Parsevals theorem of energy conservation:

|g(t)|2 dt =1

2

|G()|2 d

Selected Fourier Transforms

g(t) G()

1 (DC level) 2()

u(t) (unit step) () +1

j

ej0t 2( 0)

cos0t [( 0) + ( + 0)]

sin0t

j[( 0) ( + 0)]

n=

(t nT ) (impulse train)2

T

m=

(

2m

T

)

g(t ) (time shift) ej G()

g(at) (scale in time)1

|a|G(a

)

ej0t g(t) G( 0) (frequency shift)

g1(t) g2(t) (convolution) G1()G2() (multiplication)

g1(t) g2(t) (multiplication)1

2G1() G2() (convolution)

Duality: If g(t) transforms to p(), then p(t) transforms to 2g().

Symmetry: If g(t) is real, then G() = G() ( means complex conjugate).

If g(t) is real and even, then G() is real and even.

If g(t) is real and odd, then G() is imaginary and odd.

z-Transforms

The z-transform of a discrete-time causal sequence

g[n] (defined for n = 0, 1, 2, . . . ) is given by:

G(z) =

n=0

g[n] zn

Selected z-Transforms

g[n], (n 0) G(z)

[n] (unit pulse) 1

[nm] zm

1 (unit step)z

z 1

n (unit ramp)z

(z 1)2

rnz

z r

n rnr z

(z r)2

sin(0n)z sin0

z2 2z cos0 + 1

cos(0n)z2 z cos0

z2 2z cos0 + 1

rn sin0nz r sin0

z2 2z r cos0 + r2

rn cos0nz2 z r cos0

z2 2z r cos0 + r2

rn g[n] G(r1z

)

g[n+ 1] z G(z) zg(0)

g[n 1] z1G(z) + g(1)

Final Value Theorem:

limn

g[n] = limz1

(z1)G(z) (discrete-time)

Communications Theory

Amplitude Modulation An amplitude modu-

lated (AM) signal can be expressed as:

xc(t) = Ac [1 + amn(t)] cos (2fct)

Angle Modulation The general angle-modulated

signal is given by:

xc(t) = Ac cos (2fct+ (t))

For PM: (t) = kpm(t)

For FM: (t) = kp

tm() d

= 2fd

tm() d

Angle Modulation

(t) = sin (2fmt)

the angle-modulated signal is:

xc(t) = Ac

n=

Jn () cos [2 (fc + nfm) t]

where Jn () =

Jn () , for n even

Jn () for n odd

The bandwidth is given by Carsons Rule:

B 2 ( + 1) fm

For a FM modulator withm(t) = A cos (2fmt),

= fdA/fm

Carsons Rule For Arbitrary FM Signals:

B 2 (D + 1)W

where W is the bandwidth of the message signal

m(t), and the deviation ratio is

D =peak frequency deviation

W

C = B log2 (1 + S/N) bit/s Q = wr/BW = wrL/R = wrCR