Digital Signal Processing

By Steven W. Smith, Ph.D.

- 1: The Breadth and Depth of DSP
- 2: Statistics, Probability and Noise
- 3: ADC and DAC
- 4: DSP Software
- 5: Linear Systems
- 6: Convolution
- 7: Properties of Convolution
- 8: The Discrete Fourier Transform
- 9: Applications of the DFT
- 10: Fourier Transform Properties
- 11: Fourier Transform Pairs
- 12: The Fast Fourier Transform
- 13: Continuous Signal Processing
- 14: Introduction to Digital Filters
- 15: Moving Average Filters
- 16: Windowed-Sinc Filters
- 17: Custom Filters
- 18: FFT Convolution
- 19: Recursive Filters
- 20: Chebyshev Filters
- 21: Filter Comparison
- 22: Audio Processing
- 23: Image Formation & Display
- 24: Linear Image Processing
- 25: Special Imaging Techniques
- 26: Neural Networks (and more!)
- 27: Data Compression
- 28: Digital Signal Processors
- 29: Getting Started with DSPs
- 30: Complex Numbers
- 31: The Complex Fourier Transform
- 32: The Laplace Transform
- 33: The z-Transform
- 34: Explaining Benford's Law

Your laser printer will thank you!

Linearity of the Fourier Transform

The Fourier Transform is *linear,* that is, it possesses the properties of
*homogeneity* and *additivity*. This is true for all four members of the Fourier
transform family (Fourier transform, Fourier Series, DFT, and DTFT).

Figure 10-1 provides an example of how homogeneity is a property of the
Fourier transform. Figure (a) shows an arbitrary time domain signal, with the
corresponding frequency spectrum shown in (b). We will call these two signals:
*x*[ ] and *X*[ ], respectively. *Homogeneity* means that a change in amplitude in
one domain produces an identical change in amplitude in the other domain.
This should make intuitive sense: when the amplitude of a time domain
waveform is changed, the amplitude of the sine and cosine waves making up
that waveform must also change by an equal amount.

In mathematical form, if *x*[ ] and *X*[ ] are a Fourier Transform pair, then *kx*[ ] and *kX*[ ] are also a Fourier Transform pair, for any constant *k*. If the frequency domain is represented in *rectangular* notation, *kX*[ ] means that both the real part and the imaginary part are multiplied by *k*. If the frequency domain is
represented in *polar* notation, *kX*[ ] means that the magnitude is multiplied by
*k*, while the phase remains unchanged.

*Additivity *of the Fourier transform means that *addition* in one domain
corresponds to *addition* in the other domain. An example of this is shown in
Fig. 10-2. In this illustration, (a) and (b) are signals in the time domain called *x*_{1}[*n*] and *x*_{2}[*n*], respectively. Adding these signals produces a third time domain signal called *x*_{3}[*n*], shown in (c). Each of these three signals has a frequency
spectrum consisting of a real and an imaginary part, shown in (d) through (i).
Since the two time domain signals *add* to produce the third time domain signal,
the two corresponding spectra *add* to produce the third spectrum. Frequency
spectra are added in rectangular notation by adding the real parts to the real
parts and the imaginary parts to the imaginary parts. If: *x*_{1}[*n*] + *x*_{2}[*n*] = *x*_{3}[*n*],
then: *ReX*_{1}[*f*] + *ReX*_{2}[*f*] = *ReX*_{3}[*f*] and *ImX*_{1}[*f*] + *ImX*_{2}[*f*] = *ImX*_{3}[*f*]. Think of
this in terms of cosine and sine waves. All the cosine waves add (the real parts)
and all the sine waves add (the imaginary parts) with no interaction between the
two.

Frequency spectra in polar form cannot be directly added; they must be converted into rectangular notation, added, and then reconverted back to

polar form. This can also be understood in terms of how sinusoids behave.
Imagine adding two sinusoids having the same frequency, but with different
amplitudes (*A*_{1} and *A*_{2}) and phases (φ_{1} and φ_{2}). If the two phases happen to
be same (φ_{1} = φ_{2}), the amplitudes will add (*A*_{1} + *A*_{2}) when the sinusoids are
added. However, if the two phases happen to be exactly opposite (φ_{1} = -φ_{2}),
the amplitudes will *subtract* (*A*_{1} - *A*_{2}) when the sinusoids are added. The point
is, when sinusoids (or spectra) are in polar form, they *cannot* be added by
simply adding the magnitudes and phases.

In spite of being linear, the Fourier transform is *not* shift invariant. In other
words, a shift in the time domain *does not* correspond to a shift in the frequency
domain. This is the topic of the next section.