Properties of Fourier Representation
Multiplication property
The multiplication property defines the Fourier representation of a product of time-domain signals. We begin by considering the product of nonperiodic continuous-time signals. If x(t) and z(t) are nonperiodic signals, then we wish to express the Fourier Transform of the product y(t) = x(t)z(t) in terms of the Fourier Transform of x(t) and z(t). We represent x(t) and z(t) in terms of their respective FT’s as:
x(t) = 1/2pi ∫X(jv) e^jvt dv
And,
z(t) = 1/2pi ∫Z(jn) e^int dn.
The product term, y(t), may be written in the form:
y(t) = 1/(2pi)^2 ∫∫ X(jv) Z(jn) e^i(n+v)t dndv.
The inner integral over v represents the convolution of Z(jw) and X(jw), while the outer integral over w is of the form of the Fourier representation for y(t). Therefore, we identify this convolution, scaled by 1/(2pi), as Y(jw); that is:
y(t) = x(t) z(t) ←-> Y(jw) = 1/2pi X(jw) * Z(jw). Where, X(jw) * Z(jw) = ∫X(jv) Z(j(w-v) dv.
Multiplication of two signals in the time domain corresponds to the convolution of their FTs in the frequency domain and multiplication by the factor 1/(2pi). Similarly, if x[n] and z[n] are discrete-time nonperiodic signals, then the DTFT of the product y[n]=x[n]z[n] is given by the convolution of their DTFTs and multiplication by 1/(2pi); that is:
Y[n] = x[n] z[n] ←→ Y(e^iΩ) = 1/2pi X(e^iΩ) * Z(e^iΩ)
Where, * denotes periodic convolution.
The multiplication property enables us to study the effects of truncating a time-domain signal on its frequency-domain representation. The process of shortening a signal is known as windowing as it corresponds to viewing the signal through a window. The portion of the signal that is not visible through the window is shortened or assumed to be zero.
The windowing operation is represented mathematically by multiplying the signal, let us say x(t), by a window function w(t) that is zero outside the time range of interest. Denoting the windowed signal by y(t), we have y(t) = x(t)w(t).
Scaling properties
The scaling effect may be experienced by playing a recorded sound at a speed different from that at which it was recorded. If we play the sound back at a higher speed, corresponding to a > 1, we compress the time signal. Inverse scaling in the frequency domain expands the Fourier representation over a broader frequency band and explains the increase in the perceived pitch of the sound.
Contrariwise, playing the sound back at a slower speed corresponds to expanding the time signal, since a < 1. The inverse scaling in the frequency domain compresses the Fourier representation and explains the decrease in the perceived pitch of the sound.
Parseval relationships
The Parseval relationships state that the energy or power in the time-domain representation of a signal is equal to the energy or power in the frequency-domain representation. Hence, energy and power are conserved in the Fourier representation. We derive this result for the Fourier Transform and simply state it for the other three cases. The energy in a continuous-time nonperiodic signal is:
W = ∫|x(t)|^2 dt.
Where it is assumed that x(t) may be complex-valued in general.
Hence, the energy in the time-domain representation of the signal is equal to the energy and the frequency-domain representation, normalized by 2w. The quantity |X(jw|^2 plotted against is termed the energy spectrum of the signal.
The energy or power in the time-domain representation is equal to the energy or power in the frequency-domain representation. Energy is used for nonperiodic time-domain signals, while power applies to periodic time-domain signals. As we know that power is defined as the integral or sum of the magnitude squared over one period, and by the length of the period. The power or energy spectrum of a signal is defined as the square of the magnitude spectrum. These relationships indicate how the power or energy in the signal is distributed as a function of frequency.
Duality property of Fourier Transform
An impulse in time transforms to a constant in frequency, while a constant in time transforms to an impulse in frequency. We have also observed symmetries in Fourier representation properties: Convolution in one domain corresponds to modulation in the other domain, differentiation in one domain corresponds to multiplication by the independent variable in the other domain.
These symmetries are a consequence of the symmetry in the definitions of time- and frequency-domain representations. If we are careful, we may interchange time and frequency. This interchangeability property is termed duality.
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