## 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:

*reconstruction*of from its projections onto the coordinate axes is then the

*vector sum of the projections*:

*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:

*orthogonal*. Since they are also unit length, , we say that the coordinate vectors are

*orthonormal*.

### Changing Coordinates

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*.

#### An Example of Changing Coordinates in 2D

As a simple example, let's pick the following pair of new coordinate vectors in 2D:*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}

### Projection onto Linearly Dependent Vectors

Now consider another example:*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 ( ).

### Projection onto Non-Orthogonal Vectors

Consider this example:*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.

### General Conditions

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:**A set of vectors is said to form a

*vector space*if, given any two members and from the set, the vectors and are also in the set, where is any scalar.

**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

**Corollary:**The set of all linear combinations of

*real*vectors , using real scalars , form a vector space.

**Definition:**The set of all linear combinations of a set of

*complex*vectors from , using complex scalars, is called a

*complex vector space*of dimension .

**Definition:**The set of all linear combinations of a set of

*real*vectors from , using real scalars, is called a

*real vector space*of dimension .

**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

**Definition:**The vector space spanned by a set of vectors from is called an -dimensional

*subspace*of .

**Definition:**A vector is said to be

*linearly dependent*on a set of vectors , , if can be expressed as a linear combination of those vectors. 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 :

*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 [47]).

**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.)

### Signal/Vector Reconstruction from Projections

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

### Gram-Schmidt Orthogonalization

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 .

**Theorem:**Given a set of linearly independent vectors from , we can construct an

*orthonormal*set which are linear combinations of the original set and which span the same space.

*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 .
- Define
as minus the projection of
onto
:
- Set
(
*i.e.*, normalize the result of the preceding step). - Define
as minus the projection of
onto
and
:
- Normalize: .
- Continue this process until has been defined.

*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. This chapter can be considered an introduction to some important concepts of

*linear algebra*. The student is invited to pursue further reading in any textbook on linear algebra, such as [47].

^{5.13}Matlab/Octave examples related to this chapter appear in Appendix I.

**Next Section:**

Signal Projection Problems

**Previous Section:**

The Inner Product