This chapter provides an introduction to the elements of geometric signal theory, including vector spaces, norms, inner products, orthogonality, projection of one signal onto another, and elementary vector space operations. First, however, we will ``get our bearings'' with respect to the DFT.
We are now in a position to have a full understanding of the transform kernel:
Signals as Vectors
For the DFT, all signals and spectra are length . A length sequence can be denoted by , , where may be real ( ) or complex ( ). We now wish to regard as a vector5.1 in an dimensional vector space. That is, each sample is regarded as a coordinate in that space. A vector is mathematically a single point in -space represented by a list of coordinates called an -tuple. (The notation means the same thing as .) It can be interpreted geometrically as an arrow in -space from the origin to the point .
We define the following as equivalent:
Under the geometric interpretation of a length signal, each sample is a coordinate in the dimensional space. Signals which are only two samples long are not terribly interesting to hear,5.2 but they are easy to plot geometrically.
Given two vectors in , say
the vector sum is defined by elementwise addition. If we denote the sum by , then we have for . We could also write for if preferred.
The vector diagram for the sum of two vectors can be found using the parallelogram rule, as shown in Fig.5.2 for , , and .
Also shown are the lighter construction lines which complete the parallelogram started by and , indicating where the endpoint of the sum lies. Since it is a parallelogram, the two construction lines are congruent to the vectors and . As a result, the vector sum is often expressed as a triangle by translating the origin of one member of the sum to the tip of the other, as shown in Fig.5.3.
In the figure, was translated to the tip of . This depicts , since `` picks up where leaves off.'' It is equally valid to translate to the tip of , because vector addition is commutative, i.e., = .
Figure 5.4 illustrates the vector difference between and . From the coordinates, we compute .
Note that the difference vector may be drawn from the tip of to the tip of rather than from the origin to the point ; this is a customary practice which emphasizes relationships among vectors, but the translation in the plot has no effect on the mathematical definition or properties of the vector. Subtraction, however, is not commutative.
To ascertain the proper orientation of the difference vector , rewrite its definition as , and then it is clear that the vector should be the sum of vectors and , hence the arrowhead is on the correct endpoint. Or remember `` points to ,'' or `` is from .''
A scalar is any constant value used as a scale factor applied to a vector. Mathematically, all of our scalars will be either real or complex numbers.5.3 For example, if denotes a vector of complex elements, and denotes a complex scalar, then
In signal processing, we think of scalar multiplication as applying some constant scale factor to a signal, i.e., multiplying each sample of the signal by the same constant number. For example, a 6 dB boost can be carried out by multiplying each sample of a signal by 2, in which case 2 is the scalar. When the scalar magnitude is greater than one, it is often called a gain factor, and when it is less than one, an attenuation.
Linear Combination of Vectors
A linear combination of vectors is a sum of scalar multiples of those vectors. That is, given a set of vectors of the same type,5.4 such as (they must have the same number of elements so they can be added), a linear combination is formed by multiplying each vector by a scalar and summing to produce a new vector of the same type:
In signal processing, we think of a linear combination as a signal mix. Thus, the output of a mixing console may be regarded as a linear combination of the input signal tracks.
A set of vectors may be called a linear vector space if it is closed under linear combinations. That is, given any two vectors and from the set, the linear combination
This section defines some useful functions of signals (vectors).
The mean of a signal (more precisely the ``sample mean'') is defined as the average value of its samples:5.5
In physics, energy (the ``ability to do work'') and work are in units of ``force times distance,'' ``mass times velocity squared,'' or other equivalent combinations of units.5.6 In digital signal processing, physical units are routinely discarded, and signals are renormalized whenever convenient. Therefore, is defined above without regard for constant scale factors such as ``wave impedance'' or the sampling interval .
Power is always in physical units of energy per unit time. It therefore makes sense to define the average signal power as the total signal energy divided by its length. We normally work with signals which are functions of time. However, if the signal happens instead to be a function of distance (e.g., samples of displacement along a vibrating string), then the ``power'' as defined here still has the interpretation of a spatial energy density. Power, in contrast, is a temporal energy density.
The variance (more precisely the sample variance) of the signal is defined as the power of the signal with its mean removed:5.7
The norm (more specifically, the norm, or Euclidean norm) of a signal is defined as the square root of its total energy:
Other Lp Norms
Since our main norm is the square root of a sum of squares,
We could equally well have chosen a normalized norm:
More generally, the (unnormalized) norm of is defined as
- : The , ``absolute value,'' or ``city block'' norm.
- : The , ``Euclidean,'' ``root energy,'' or ``least squares'' norm.
- : The , ``Chebyshev,'' ``supremum,'' ``minimax,'' or ``uniform'' norm.
There are many other possible choices of norm. To qualify as a norm on , a real-valued signal-function must satisfy the following three properties:
- , with
Mathematically, what we are working with so far is called a Banach space, which is a normed linear vector space. To summarize, we defined our vectors as any list of real or complex numbers which we interpret as coordinates in the -dimensional vector space. We also defined vector addition (§5.3) and scalar multiplication (§5.5) in the obvious way. To have a linear vector space (§5.7), it must be closed under vector addition and scalar multiplication (linear combinations). I.e., given any two vectors and from the vector space, and given any two scalars and from the field of scalars , the linear combination must also be in the space. Since we have used the field of complex numbers (or real numbers ) to define both our scalars and our vector components, we have the necessary closure properties so that any linear combination of vectors from lies in . Finally, the definition of a norm (any norm) elevates a vector space to a Banach space.
The inner product (or ``dot product'', or ``scalar product'') is an operation on two vectors which produces a scalar. Defining an inner product for a Banach space specializes it to a Hilbert space (or ``inner product space''). There are many examples of Hilbert spaces, but we will only need for this book (complex length vectors, and complex scalars).
The inner product between (complex) -vectors and is defined by5.9
Note that the inner product takes to . That is, two length complex vectors are mapped to a complex scalar.
Any function of a vector (which we may call an operator on ) is said to be linear if for all and , and for all scalars and in ,
The inner product is linear in its first argument, i.e., for all , and for all ,
The inner product is also additive in its second argument, i.e.,
The inner product is strictly linear in its second argument with respect to real scalars and :
Since the inner product is linear in both of its arguments for real scalars, it may be called a bilinear operator in that context.
The Cauchy-Schwarz Inequality (or ``Schwarz Inequality'') states that for all and , we have
We can quickly show this for real vectors , , as follows: If either or is zero, the inequality holds (as equality). Assuming both are nonzero, let's scale them to unit-length by defining the normalized vectors , , which are unit-length vectors lying on the ``unit ball'' in (a hypersphere of radius ). We have
The triangle inequality states that the length of any side of a triangle is less than or equal to the sum of the lengths of the other two sides, with equality occurring only when the triangle degenerates to a line. In , this becomes
A useful variation on the triangle inequality is that the length of any side of a triangle is greater than the absolute difference of the lengths of the other two sides:
Proof: By the triangle inequality,
Interchanging and establishes the absolute value on the right-hand side.
The Cauchy-Schwarz Inequality can be written
Note that if and are real and orthogonal, the cosine of the angle between them is zero. In plane geometry (), the angle between two perpendicular lines is , and , as expected. More generally, orthogonality corresponds to the fact that two vectors in -space intersect at a right angle and are thus perpendicular geometrically.
Let and , as shown in Fig.5.8.
The inner product is . This shows that the vectors are orthogonal. As marked in the figure, the lines intersect at a right angle and are therefore perpendicular.
This relationship generalizes to dimensions, as we can easily show:
If , then and Eq.(5.1) holds in dimensions.
Note that the converse is not true in . That is, does not imply in . For a counterexample, consider , , in which case
For real vectors , the Pythagorean theorem Eq.(5.1) holds if and only if the vectors are orthogonal. To see this, note that, from Eq.(5.2), when the Pythagorean theorem holds, either or is zero, or is zero or purely imaginary, by property 1 of norms (see §5.8.2). If the inner product cannot be imaginary, it must be zero.
Note that we also have an alternate version of the Pythagorean theorem:
The orthogonal projection (or simply ``projection'') of onto is defined by
Motivation: The basic idea of orthogonal projection of onto is to ``drop a perpendicular'' from onto to define a new vector along which we call the ``projection'' of onto . This is illustrated for in Fig.5.9 for and , in which case
Signal Reconstruction from Projections
We now know how to project a signal onto other signals. We now need to learn how to reconstruct a signal from its projections onto different vectors , . This will give us the inverse DFT operation (or the inverse of whatever transform we are working with).
As a simple example, consider the projection of a signal onto the rectilinear coordinate axes of . The coordinates of the projection onto the 0th coordinate axis are simply . The projection along coordinate axis has coordinates , and so on. The original signal is then clearly the vector sum of its projections onto the coordinate axes:
To make sure the previous paragraph is understood, let's look at the details for the case . We want to project an arbitrary two-sample signal onto the coordinate axes in 2D. A coordinate axis can be generated by multiplying any nonzero vector by scalars. The horizontal axis can be represented by any vector of the form , while the vertical axis can be represented by any vector of the form , . For maximum simplicity, let's choose
The projection of onto is, by definition,
Similarly, the projection of onto is
The reconstruction of from its projections onto the coordinate axes is then the vector sum of the projections:
The projection of a vector onto its coordinate axes is in some sense trivial because the very meaning of the coordinates is that they are scalars to be applied to the coordinate vectors in order to form an arbitrary vector as a linear combination of the coordinate vectors:
What's more interesting is when we project a signal onto a set of vectors other than the coordinate set. This can be viewed as a change of coordinates in . In the case of the DFT, the new vectors will be chosen to be sampled complex sinusoids.
As a simple example, let's pick the following pair of new coordinate vectors in 2D:
These happen to be the DFT sinusoids for having frequencies (``dc'') and (half the sampling rate). (The sampled complex sinusoids of the DFT reduce to real numbers only for and .) We already showed in an earlier example that these vectors are orthogonal. However, they are not orthonormal since the norm is in each case. Let's try projecting onto these vectors and seeing if we can reconstruct by summing the projections.
The projection of onto is, by definition,5.12
Similarly, the projection of onto is
The sum of these projections is then
Now consider another example:
The projections of onto these vectors are
The sum of the projections is
Something went wrong, but what? It turns out that a set of vectors can be used to reconstruct an arbitrary vector in from its projections only if they are linearly independent. In general, a set of vectors is linearly independent if none of them can be expressed as a linear combination of the others in the set. What this means intuitively is that they must ``point in different directions'' in -space. In this example so that they lie along the same line in -space. As a result, they are linearly dependent: one is a linear combination of the other ( ).
Consider this example:
These point in different directions, but they are not orthogonal. What happens now? The projections are
The sum of the projections is
So, even though the vectors are linearly independent, the sum of projections onto them does not reconstruct the original vector. Since the sum of projections worked in the orthogonal case, and since orthogonality implies linear independence, we might conjecture at this point that the sum of projections onto a set of vectors will reconstruct the original vector only when the vector set is orthogonal, and this is true, as we will show.
It turns out that one can apply an orthogonalizing process, called Gram-Schmidt orthogonalization to any linearly independent vectors in so as to form an orthogonal set which will always work. This will be derived in Section 5.10.4.
Obviously, there must be at least vectors in the set. Otherwise, there would be too few degrees of freedom to represent an arbitrary . That is, given the coordinates of (which are scale factors relative to the coordinate vectors in ), we have to find at least coefficients of projection (which we may think of as coordinates relative to new coordinate vectors ). If we compute only coefficients, then we would be mapping a set of complex numbers to numbers. Such a mapping cannot be invertible in general. It also turns out linearly independent vectors is always sufficient. The next section will summarize the general results along these lines.
This section summarizes and extends the above derivations in a more formal manner (following portions of chapter 4 of ). In particular, we establish that the sum of projections of onto vectors will give back the original vector whenever the set is an orthogonal basis for .
Definition: The set of all -dimensional complex vectors is denoted . That is, consists of all vectors defined as a list of complex numbers .
Theorem: is a vector space under elementwise addition and multiplication by complex scalars.
Proof: This is a special case of the following more general theorem.
Theorem: Let be an integer greater than 0. Then the set of all linear combinations of vectors from forms a vector space under elementwise addition and multiplication by complex scalars.
Proof: Let the original set of vectors be denoted . Form
which is yet another linear combination of the original vectors (since complex numbers are closed under addition). Since we have shown that scalar multiples and vector sums of linear combinations of the original vectors from are also linear combinations of those same original vectors from , we have that the defining properties of a vector space are satisfied.
Corollary: The set of all linear combinations of real vectors , using real scalars , form a vector space.
Definition: If a vector space consists of the set of all linear combinations of a finite set of vectors , then those vectors are said to span the space.
Example: The coordinate vectors in span since every vector can be expressed as a linear combination of the coordinate vectors as
Thus, is linearly dependent on if there exist scalars such that . Note that the zero vector is linearly dependent on every collection of vectors.
Theorem: (i) If span a vector space, and if one of them, say , is linearly dependent on the others, then the same vector space is spanned by the set obtained by omitting from the original set. (ii) If span a vector space, we can always select from these a linearly independent set that spans the same space.
Proof: Any in the space can be represented as a linear combination of the vectors . By expressing as a linear combination of the other vectors in the set, the linear combination for becomes a linear combination of vectors other than . Thus, can be eliminated from the set, proving (i). To prove (ii), we can define a procedure for forming the required subset of the original vectors: First, assign to the set. Next, check to see if and are linearly dependent. If so (i.e., is a scalar times ), then discard ; otherwise assign it also to the new set. Next, check to see if is linearly dependent on the vectors in the new set. If it is (i.e., is some linear combination of and ) then discard it; otherwise assign it also to the new set. When this procedure terminates after processing , the new set will contain only linearly independent vectors which span the original space.
Definition: A set of linearly independent vectors which spans a vector space is called a basis for that vector space.
Definition: The set of coordinate vectors in is called the natural basis for , where the th basis vector is
Theorem: The linear combination expressing a vector in terms of basis vectors for a vector space is unique.
Proof: Suppose a vector can be expressed in two different ways as a linear combination of basis vectors :
where for at least one value of . Subtracting the two representations gives
Note that while the linear combination relative to a particular basis is unique, the choice of basis vectors is not. For example, given any basis set in , a new basis can be formed by rotating all vectors in by the same angle. In this way, an infinite number of basis sets can be generated.
As we will soon show, the DFT can be viewed as a change of coordinates from coordinates relative to the natural basis in , , to coordinates relative to the sinusoidal basis for , , where . The sinusoidal basis set for consists of length sampled complex sinusoids at frequencies . Any scaling of these vectors in by complex scale factors could also be chosen as the sinusoidal basis (i.e., any nonzero amplitude and any phase will do). However, for simplicity, we will only use unit-amplitude, zero-phase complex sinusoids as the Fourier ``frequency-domain'' basis set. To summarize this paragraph, the time-domain samples of a signal are its coordinates relative to the natural basis for , while its spectral coefficients are the coordinates of the signal relative to the sinusoidal basis for .
Theorem: Any two bases of a vector space contain the same number of vectors.
Proof: Left as an exercise (or see ).
Definition: The number of vectors in a basis for a particular space is called the dimension of the space. If the dimension is , the space is said to be an dimensional space, or -space.
In this book, we will only consider finite-dimensional vector spaces in any detail. However, the discrete-time Fourier transform (DTFT) and Fourier transform (FT) both require infinite-dimensional basis sets, because there is an infinite number of points in both the time and frequency domains. (See Appendix B for details regarding the FT and DTFT.)
We now arrive finally at the main desired result for this section:
Theorem: The projections of any vector onto any orthogonal basis set for can be summed to reconstruct exactly.
Proof: Let denote any orthogonal basis set for . Then since is in the space spanned by these vectors, we have
for some (unique) scalars . The projection of onto is equal to
Recall from the end of §5.10 above that an orthonormal set of vectors is a set of unit-length vectors that are mutually orthogonal. In other words, orthonormal vector set is just an orthogonal vector set in which each vector has been normalized to unit length .
Proof: We prove the theorem by constructing the desired orthonormal set sequentially from the original set . This procedure is known as Gram-Schmidt orthogonalization.
First, note that for all , since is linearly dependent on every vector. Therefore, .
- Set .
as minus the projection of
- Set (i.e., normalize the result of the preceding step).
as minus the projection of
- Normalize: .
- Continue this process until
has been defined.
The Gram-Schmidt orthogonalization procedure will construct an orthonormal basis from any set of linearly independent vectors. Obviously, by skipping the normalization step, we could also form simply an orthogonal basis. The key ingredient of this procedure is that each new basis vector is obtained by subtracting out the projection of the next linearly independent vector onto the vectors accepted so far into the set. We may say that each new linearly independent vector is projected onto the subspace spanned by the vectors , and any nonzero projection in that subspace is subtracted out of to make the new vector orthogonal to the entire subspace. In other words, we retain only that portion of each new vector which ``points along'' a new dimension. The first direction is arbitrary and is determined by whatever vector we choose first ( here). The next vector is forced to be orthogonal to the first. The second is forced to be orthogonal to the first two (and thus to the 2D subspace spanned by them), and so on.
Matlab/Octave examples related to this chapter appear in Appendix I.
Derivation of the Discrete Fourier Transform (DFT)
Sinusoids and Exponentials