Fourier

• To apprise the intimate relationship between the 4-Fourier’s used in practice.

• Continuous Time Fourier Series and Continuous Time Fourier Transform.

• Continuous Time Fourier Transform. • Discrete Time Fourier Transform and • Continuous Time Fourier Series and • Discrete Time Fourier Series, the latter

redesignated as Discrete Fourier Transform.

•

Brief on Fourier Representation

• There are Four Fourier representations:

Continuous time Fourier Series (CTFS): yields aperiodic discrete frequency spectraof a continuous periodic wave. Continuous time Fourier Transform (CTFT): yields aperiodic Continuous frequency spectraof an aperiodic continuous time signal.

.Brief on Fourier Representation

The remaining Two Fourier representations:Discrete time Fourier Transform (DTFT):

yields periodic and continuous frequency spectraof an aperiodic discrete signal.

Discrete Time Fourier Series (DTFS): or

Discrete Fourier Transform (DFT): yields periodic but Discrete frequency spectra

of a periodic discrete signal.

Conclusion

• A periodic time signal has a discrete frequency spectra.

• A discrete time signal displays a periodic frequency spectra.

• If the time signal is aperiodic, it’s spectra is continuous in frequency.

• The continuous signal yield an aperiodic spectra.

Important

• When we analyze a signal in time domain, we fragment the signal in time.

• When we analyze a signal in frequency domain, we fragment the signal in frequencies.

• If we analyze the signal in wavelets, we fragment the signal in the shape of mother wavelet.

CTFS_1

Any arbitrary periodic & continuous waveform in time-domain, A.x(t/T), having

amplitude = A and periodicity of T (=1/f) is decomposed in discrete frequency components nf ; where n [0:1: ].

For the sack of simplicity, the peak of the amplitude A as well as frequency f is often taken as unity.

The shape of the waveform decides the relative magnitude of frequency spectra.

The location of the ordinate decides the phase of different frequency components of the decomposed waveform.

CTFS_2

• The representation of plots of magnitude and phase of the decomposed waveform in frequencies is also termed as line spectra in magnitude and phase.

• Smooth joining of spectra points yield magnitude and phase vrs frequency plots.

• Magnitude in dB, phase in linear scale and frequency in log-scale corresponds to Bode-Plot.

• The [magnitude]2 vrs frequency spectra is called power spectrum or only spectra. Since power is scalar, phase is meaningless here.

• Quiz: Draw Bode plot for square pulse train and traingular pulse train. Interpret the results.

Interrogation on Fourier Series of a square wave

• To normalize the FS, amplitude of the square wave is taken to be /2. Duty cycle is 50%.

• Waveform is periodic in the time range ∞.

• Periodicity is Tp =1/fs.

• The resultant FS can be either x1(t) or,x2(t)

• x1(t)= [cost - cos3t/3 + cos5t/5-..].

• x2(t)= [sint + sin 3t/3 + sin5 t/5-..]

• Quiz: a. Workout Frequency plot and Bode plot• Where the ordinate should lie in either case.

CTFS_3

For the sake of ease, it is assumed that the power is delivered into a resistive load of 1 . Thus each frequency component of the waveform feeds into this load.

Total power equals summing the power contained in each frequency component. Power so calculated equals that in time domain.

See that the frequency spectra is evenly spaced discrete and is aperiodic while the time waveform is continuous and periodic.

CTFS_4The discrete but aperiodic spectra of CTFS exhibit that the magnitude decreases as the harmonics increases.

It implies that in the periodic waves, power converges to a finite quantity. Energy is infinite.

Such signals are dealt as power signals.

First few frequency components contain bulk of power.

The higher frequency components contain a negligible fraction of total power.

CTFS_5

Higher frequency components control the fineness and smoothness of a signal.

In time domain, this information is contained in rate of rise time and rate of fall time of the waveform.

Gibbs proved that no waveform with discontinuities (sudden change, as in pulse) can be reconstructed by synthesis procedure without 14% peaks at discontinuities.

CTFS: graphical representation

relationship

CONTINUOUS TIME FOURIER TRANSFORM: (CTFT):

Information are non periodic and of unknown shape. To learn the analysis, we take a single sample of a known continuous time periodic waveform A x(t/T).This sample, represented by x(t) is non-periodic and is limited to time-width, T. The time width of one time-period, T is taken.For the sake of simplicity A = 1 and T = 1 is assumed.

CTFT..

In analysis the origin is taken as center of the signal.

The Fourier Transform of this one period wide aperiodic continuous-time-wave is continuous and aperiodic in frequency domain.

Since the signal wave is aperiodic, it has zero power in the time range [-: ].

We deal such signals for ‘energy’.

CTFT…

• For window of period T, the CTFT of x(t) is the simplified and normalized version of the FS. The results obtained in FS are compatible with FT.

FT of a Pulse: a sinc fnNote that pulse width =1, has nothing to do with

periodicity. However arrows are marked for = T/2, we get the coefficients of FS with 50% duty cycle.

• .

F f( )

sin 2 f2

2 f2

1 f 4 3.9 4 F f( )

sin 2 f2

2 f2

4 3 2 1 0 1 2 3 41

0.5

0

0.5

1

1.5

2

2

F f( )

f

1/

1/T

3/T

5/T2/T

7/T

Comparison of Result: FS and FT

• The Sinc Function is the Fourier Transform of a pulse wave.

• If the pulse waveform has 50% duty cycle, it will contain First, Third and Fifth Harmonics as shown by arrows. Their respective amplitudes are 1, 1/3, 1/5,1/7 etc.

• The second, fourth, sixth or, all odd harmonics are essentially zero.

• The results are compatible with the results of FS .

• What if the duty cycle is not 50%? We discuss

Fourier Transform of Gated Cosine wave

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-1

-0.5

0

0.5

1

t

x(t

)

-20 -15 -10 -5 0 5 10 15 20-0.5

0

0.5

1

1.5

f

X(f

)

gated cosine

FourierTransform

FT of gated wave..

• In the above slide, 20 cycles of cosine wave passes through a pulse type gate function in the duration -1 to 1 seconds.

• In time domain, the cosine wave function multiplies with the gate function.

• It corresponds to convolution in frequency domain of FT of cosine wave with FT of pulse function.

FT of a Pulse: a sinc fn.

• .

F f( )

T sin 2 fT

2

2 fT

2

1 f 4 3.9 4 F f( )

sin 2 f2

2 f2

4 3 2 1 0 1 2 3 41

0.5

0

0.5

1

1.5

2

2

F f( )

f

1/

CTFT…

The Inverse of this Fourier Transform (ICTFT) returns result in time domain.

Proportional Fourier Series components can be found by drawing ordinates at the frequencies [0:1:]/T; where T is the time period of the signal.

Likewise, by smoothening and normali-zing the ordinate points of FS of x(t), one can arrive at plots of CTFT of x(t).

CTFT

While the shape of the signal decides the magnitude characteristics, the phase depends on the position of the ordinate.Fourier Transform is more flexible to use compared to Fourier Series.CTFT is closely related to Laplace-Transform (LT) and linear time invariant differential equations.The frequency response pertains to steady state time response.

CTFT….

note

Discrete Time Fourier Transform [DTFT]

It is an extension of CTFT.

One cycle of periodic signal x(t) of periodicity Tp=(1/fp )(now non periodic), is continuously sampled by an ideal impulse switch at an interval Ts (=1/fs).

The time sampled output is modeled as

xs(nTs) = x(t). (t – nTs); [- ≤ n ≤ ].

DTFT

• In frequency domain, since• x(t) = X(f) and • (t - n/fs) = (f - nfs).

• The xs(nTs) = x(t). (t – nTs); • = X(f)* (f - nfs): range [- ≤ n ≤ ].

• Thus in DTFT, the spectra of “aperiodic” x(t) • repeats • at every nfs:range [- ≤ n ≤ ]. repeats

DTFT• We define Digital Frequency FD = f/fs ½.

• The principal range is a normalized frequency range lie between [-:] rad/sec or, [-0.5:0.5] Hz.

• As per Nyquist, should FD lie within the principal range, the sampled output will be alias free.

• In DTFT, the principal range repeats every 1/fs.

• If FD < 0.5; signal lie within principal range;

• case of over sampling, no aliasing.

• If FD = 0.5; it is critical, Nyquist minimum rate.

• If FD >1/2; signal extends beyond principal range; case of under sampling. Aliased signals

generated. • .

DTFT…

The DTFT is Fourier Transform of a Discrete Time signal x[n]; is concerned with the

(a) sampling function, that in ideal case is the impulse train in frequency (f-k/T) and (b) Fourier transform X(f) of the signal x(t).The result is termed as Discrete Time Fourier Transform or, DTFT.

• The spectra of x[n] is continuous in the principal range and repeats after every fs in the frequency range of [- :].

DTFT….

In short, the frequency spectra of CTFT repeats in DTFT after every 2 rad in the range [-:].

Alternatively

The Digital Frequency Plot repeats after every cycle in the range [-FD/2:FD/2] and normalized to [-0.5:0.5] on frequency scale of Hz.

DTFT….

DTFT after calculation, turns out to be a complex quantity. It can be expressed in either cartian or, polar form.

DTFT relates to z-transform and linear difference equations with constant coefficients [ D = e- j = z-1] .

Interpolation and extrapolation are feasible.

Refer: Ambarder: PP482:484

DTFT…..

Note with care

DTFS

It is an extension of CTFS.

A time domain signal x(t) of periodicity T is sampled at a regular interval Ts (=1/fs) where NTs=T and N is the number of samples per cycle is an integer.

The sampled signal is denoted by x[nT ] or, simply by x[n].

The frequency spectra and time-wave, both are discrete.

The discrete-time waveform as well as discrete frequency spectra, both are periodic.

DTFS

CTFS has aperiodic discrete spectra in the entire frequency range; while in DTFS, a limited spectra is copied and pasted after every nfs :n is an integer in the range [].Being a finite and discrete length of series between “n and n-1”, DTFS has no convergence issue. Most properties of CTFS/CTFT/DTFT are alike.The DTFS and DFT are related to each other by the relation X[k] = NCk.

DTFS and DFT

• The important features of DFT are:• One-to-one correspondence between x[n] and

X[k].• Fast Fourier Transform (FFT) is available for

calculations.• DTFS is related to DTFT in the same way as

CTFS is related to CTFT. • Due to its finite discrete length N in time and

same in frequency domains, DFT is most appropriate Fourier representation for digital simulation.

DTFS