Continuous Time Signals

Continuous Time Signals

All signals in nature are in continuous time

)(tx

t

From Discrete Time to Continuous Time

A continuous time signals can be viewed as the limit of a discrete time signal with sampling interval 0ST

)(tx

0ST

)( SnTx

From Discrete Time FT (DTFT) …

We saw the DTFT of a discrete time signal

deXnTx

enTxX

njDTFTS

n

njSDTFT

)(2

1)(

)()(

Substitute and obtain:SS

TFF

F 22

2/

2/

2

2

)2(22

1)(

)(2

S

S

S

S

F

F

nTFjSDTFTSS

nS

nTFjSSDTFTS

dFeTFXTnTx

TenTxTFXT

… to Continuous Time FT

Now take the limit

so that discrete time -> cont. time

0ST

dFeFXFXIFTtx

dtetxtxFTFX

Ftj

Ftj

2

2

)()()(

)()()(

tnTS

SF

dtTS ......

Then we obtain the Fourier Transform

sampling freq -> infinity

sum -> integral

Fourier Transform

We want to represent a signal in terms of its frequency components.

Define: Fourier Transform (FT)

dFeFXFXIFTtx

dtetxtxFTFX

Ftj

Ftj

2

2

)()()(

)()()(

Example of a Fourier Transform

Take a Rectangular Pulse

t2/0T 2/0T

1

0

)(T

trecttx

00

0

2/22/22/

2/

2

sincsin

2)(

000

0

F

FT

F

FT

Fj

eedteFX

FTjFTjT

T

Ftj

Example of a Fourier Transform

t2/0T 2/0T

1

0

)(T

trecttx

-50 -40 -30 -20 -10 0 10 20 30 40 50-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

F(Hz)

-50 -40 -30 -20 -10 0 10 20 30 40 50-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

F(Hz)

00sinc)(

F

FTFX

t2/0T 2/0T

1

sec1.00 T

sec10 T

HzF 10

HzF 100

00 /1 TF

Properties of the FT: 1. Symmetry

If the signal is real, then its FT is symmetric as)(tx

)()( * FXFX

since

)()()()( *

*

22 FXdtetxdtetxFX FtjFtj

Example: just verify the previous example

Symmetry of the FT

|)(||)(| FXFX

)()( FXFX

F

F

Magnitude has “even”

symmetry

Phase has “odd”

symmetry

Properties of the FT: 2. Time Shift

)()( 020 FXettxFT Ftj

since

')'( )'(let

)()(

'220

200

0 dtetxettt

dtettxttxFT

FtjFtj

Ftj

In other words a time shift affects the phase, not the magnitude

Bandwidth of a Baseband Signal

• A Baseband Signal has all frequency components at the low frequencies, around F=0 Hz;

• Bandwidth: the frequency interval where most of the frequency components are.

F

|)(| FX

BB

What does it mean?

If you take the signal at two different times and

with then

t tt

Bt /1 )()( ttxtx

)()(

)()(

22

22

txdFeeFX

dFeeFXttx

B

B

tFjFtj

tFjFtj

1|| tBtF1 since

For Example:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

10

20

30

40

50

60

70

F (kHz)0 10 20 30 40 50 60 70 80 90 100

-1.5

-1

-0.5

0

0.5

1

t (msec)

31.2 31.4 31.6 31.8 32 32.2

-0.6

-0.4

-0.2

0

0.2

0.4

t (msec)

zoom

sec1.0 m

)(tx |)(| FX

)(kHzFkHzB 1

sec1/1 mB

samples spaced by less than 0.1msec are fairly

close to each other

Computation of the Fourier Transform

• Whatever we do, physical signals are in continuous time and, as we have seen, they are described by the FT;

• The FT can be computed in one of two ways:

1. Analytical: if we have an expression of the signal (like in the example);

2. Numerical: by approximation using the Fast Fourier Transform (FFT).

Fourier Transform and FFT

Consider a signal of a finite duration

00),( Tttx

with Bandwidth . Then we can approximate, by simple arguments,

B

1

0

2

0

2 )()()()(0 M

nS

FnTjS

TFtj TenTxdtetxtxFTFX S

where (say at least an order of magnitude smaller)B

TS1

)/( 0 STTroundM

Fourier Transform and FFT

Using the FFT:

• Take an even integer . Then compute the N point FFT of the sampled data, padded with zeros:

• Assign the frequencies:

12

,...,0 ],[

1,...,2

],[

NkkXT

N

FkX

NkkNXT

N

FkX

SS

SS

1,...,0,0,...,0,0),1(),...,0(][ NkMxxFFTkX

positive frequencies

negative frequencies

MN

Example

Take a sinusoid with frequency

and length

Let the sampling frequency be

kHzF 100

sec50 mT

kHzFS 200

Example

X=fft(x, N);F=(-N/2:N/2 -1)*Fs/N;plot(F,fftshift(20*log10(abs(X))))

-80 -60 -40 -20 0 20 40 60 80 100-100

-50

0

50

kHz

dB

)(kHzF

dBFX |)(|

Example (Zoom in at the Peak)

)(kHzF

dBFX |)(|

7 8 9 10 11 12 13 14

-80

-60

-40

-20

0

kHz

dB

Max at F=10kHz

Sidelobes due to finite data length

Complex Signals

All signals in nature are real. There is not such as a thing as “complex” signal.

However in many cases we are interested in processing and transmitting “pairs” of signals. We can analyze them “as if” they were just one complex signal:

)()()( tjbtatx

)(tx)(ta

)(tb

jReal Signals

Complex Signal

Amplitude Modulation: Real Signal

You want to transmit a signal over a medium (air, water, space, cable…). You need to “modulate it” by a carrier frequency:

)(ts

)2cos( tFC

)2cos()()( tFtstx CAM

0 50 100 150 200 250 300 350 400 450 500-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400 4500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Amplitude Modulation: Complex Signal

However most of the times the signal we modulate is Complex

Re{.}

Notice now that the modulated signal is real and it contains both signals a(t) and b(t).

)2sin()(

)2cos()()(

tFtb

tFtatx

C

C)()()( tjbtats

tFjj Cee 2

FT of Modulated Signal

See the different steps: ( )s tRe{.}

( )x t ( )x t

tFjj Cee 2

)()()()()( )(222C

jtFFjjFtjtFjj FFSedtetsedteetsetxFTFX CC

)(2

1)(

2

1)(Re)( * txtxtxtx

)()()()()( **

*

22**C

jFtjFtj FFSeFXdtetxdtetxtxFT

FT of Modulated Signal

Put things together:

( )s tRe{.}

( )x t ( )x t

|)(| FS

FB

| ( ) |X F

FCFCF

Usually CFB

tFjj Cee 2

)(2

1)(

2

1)( *

Cj

Cj FFSeFFSeFX