2. Continuous Fourier Transform Flashcards
What is a periodic continuous-time signal?
How is period T defined?
Signals that are defined for all values of t ∈ R are called continuous-time signals or analog signals.
Signal xT (t) is said to be periodic if there exists a value T != 0 such that
xT (t) = xT (t + T), ∀t ∈ R
If T is the smallest value larger than 0 for wich this equation holds, it is called the period of the signal.
(p. 18)
Give the definition of the Fourier series.
The Fourier series is a dedicated signal transform to convert periodic continuous-time signals from the time domain into the frequency domain.
(p. 19)
What is the meaning of the Cartesian form of the Fourier series?
What do fundamental frequency and harmonics mean?
The formulas defining the Cartesian form of the Fourier series can be found in the formulary, so you do not have to memorize them.
A periodic continuous-time signal xT(t) of finite period T can be expressed as an infinite sum of mutually orthogonal sine wave functions cos(2πfnt) and sin(2πfnt) whose frequency fn = n/T is an integer multiple of the so-called fundamental frequency 1/T of the signal.
Multiples n/T of the fundamental frequency are called harmonics. Coefficients a0, a1, b1, a2, b2, … are referred to as the Fourier coefficients of xT(t). They indicate the signal strength of the different harmonic components.
(p.19)
What is the meaning of the complex form of the Fourier series?
Which interpretation can be given to the magnitude and the phase spectrum of xT(t)?
The formulas defining the complex form of the Fourier series can be found in the formulary.
A disadvantage of the Fourier series expressed in the cartesian form is that for each frequency n/T two coefficients are involved. The complex form combines two such parameters belonging together into one single, complex variable. One obtains the formulas on p. 20 which leads to the complex or exponential form of the Fourier series.
The set of coefficients cn is called the (frequency) spectrum of xT (t). They are usually expressed in polar coordinates, i.e. as cn = |cn| · e^j∠cn . The absolute value of the coefficients |cn| is referred to as the magnitude (or amplitude) spectrum of xT (t), whereas the set of angles ∠cn is called the phase spectrum of xT (t). Magnitude and phase are typically plotted as a function of n.
The following interpretation can be given to |cn| and ∠cn.
Magnitude |cn| = An/2 reveals the amplitude of e^(j2πnt/T) , i.e. |cn| is the strength of the n-th harmonic component of xT (t). In other words, |cn| tells how much of frequency n/T there is in signal xT (t).
Angle ∠cn = ϕn on the other hand, indicates the phase of the n-th harmonic component of xT (t). In other words, |cm| tells how much of frequency m/T there is in signal xT (t).
(p. 20-22)
What do the magnitude and the phase spectrum of a real-valued periodic continuoustime signal look like?
Which symmetry can be observed?
Zie grafieken op p. 21.
Magnitude vertoont even symmetrie en phase vertoont oneven symmetrie, bij n = 0 is de fase dus altijd 0.
Door deze symmetrie kan je de negatieve frequenties negeren want wanneer je de positieve hebt, kan je de negatieve ook afleiden.
De x-as kan alleen gehele getallen bevatten, hierdoor is het spectrum altijd discreet. Wanneer je de x-as in Hz wil uitdrukken, hoef je enkel de gehele getallen te delen door de periode (n/T).
Give the definition of the continuous Fourier transform.
The continuous Fourier transform is a dedicated signal transform to convert aperiodic continuous-time signals into the frequency domain.
What will (a)periodicity in the time domain lead to in the frequency domain? (Continuous Fourier transform/Fourier series)
Aperiodicity in the time domain leads to continuity in the frequency domain, while periodic repetition in the time domain gives rise to a discrete spectrum.
The continuous Fourier transform can be expressed in terms of ω or in terms of f. Both variants are used.
The formulas defining the continuous Fourier transform can be found in the formulary.
p. 24
What is the real part, the imaginary part, the magnitude and the phase of X(f)?
Formulas to convert between the Cartesian and the polar representation can be found in the formulary.
X(f) = Xr(f) + jXi(f) = |X(f)|e^j∠X(f)
Xr(f) , Xi(f), |X(f)| and ∠X(f) are real-valued functions of f representing the real part, the imaginary part, the magnitude (also called amplitude), and the phase of X(f), respectively.
Xr(f) = |X(f)| cos(∠X(f)) –> |X(f)| = (Xr^2(f) + Xi^2(f))^1/2
Xi(f) = |X(f)|sin(∠X(f)) ∠X(f) = atan Xi(f)/Xr(f) + k · 180°
where k is an even integer if Xr(f) > 0 and k is odd if Xr(f) < 0. If Xr(f) = 0, the phase is set to ∠X(f) = sign(Xi(f)) · 90°.
(p. 25)
What is the physical meaning of |X(f)| and ∠X(f)?
The magnitude spectrum |X(f)| reveals how much of frequency f there is in signal x(t) relative to other frequencies. Hence it indicates which frequency components x(t) consists of and which not.
The physical meaning of the phase spectrum ∠X(f) is less clear, but that it is an essential part of the Fourier transform as ∠X(f) is needed to reconstruct the time-domain signal x(t) from the above equation : it is generally impossible to retrieve x(t) solely from the magnitude spectrum |X(f)|.
(p. 25)
What do the magnitude and the phase of the continuous Fourier transform of a real-valued signal look like?
Which symmetry can be observed?
See figure 2.3
The magnitude spectrum is an even function of f.
The phase spectrum is an odd function of f.
If the frequency spectrum does not show symmetry, one must conclude that the time-domain signal is complex-valued.
The spectra are continuous functions.
(p. 26)
What is the time-shifting property of the continuous Fourier transform?
Explain in a few words.
The formula can be found in the formulary.
If x(t) ←→ X(f) is a continuous Fourier transform pair, then also
x(t − t0) ←→ X(f) · e^−j*2*π*f*t0
is a continuous Fourier transform pair. Hence, time delaying a signal by t0 leaves the magnitude spectrum |X(f)e^−j2πf t0| = |X(f)| unchanged, but adds an extra phase term −2πf t0 to ∠X(f). Remark that the extra phase term rises linearly with f and t0. Because of the linear dependence on f, the term −2πf t0 is called a linear phase shift.
(p. 27)
What is a linear phase shift?
Time shifting:
If x(t) ←→ X(f) is a continuous Fourier transform pair, then also
x(t − t0) ←→ X(f) · e^−j*2*π*f*t0
is a continuous Fourier transform pair. Hence, time delaying a signal by t0 leaves the magnitude spectrum |X(f)e^−j2πf t0| = |X(f)| unchanged, but adds an extra phase term −2πf t0 to ∠X(f). Remark that the extra phase term rises linearly with f and t0. Because of the linear dependence on f, the term −2πf t0 is called a linear phase shift.
(p. 27)
What is a continuous convolution?
Explain in words how it can be calculated.
The formula can be found in the formulary.
The ⋆-sign symbolizes the (continuous) convolution operator, which is defined as:
formula
Apparently, the convolution of two signals comes down to time inverting one of the signals (change y(τ ) into y(−τ )), shifting this time-inverted signal to the right by a time lag t (leading to y(t − τ )), and integrating the product of the time-inverted, time-shifted signal y(t − τ ) and the other signal x(τ ) over the entire time axis, i.e. for all τ. This process has to be repeated for each time lag t.
-> see figure 2.6
(p. 28-29)
What does the convolution theorem state?
The formula can be found in the formulary.
If x(t) ←→ X(f) and y(t) ←→ Y (f) are continuous Fourier transform pairs, then
x(t) ⋆ y(t) ←→ X(f) · Y (f),
x(t) · y(t) ←→ X(f) ⋆ Y (f)
are also continuous Fourier transform pairs.
y(t) = h(t) ⋆ x(t) The convolution theorem proves that y(t) can also be straightforwardly computed in the frequency domain as
Y(f) = H(f) · X(f) where H(f) is the so-called frequency response14 of the system, being the continuous Fourier transform of the impulse response h(t).
(p. 28-29)
