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Fundamentals of Multirate Signal Processing

Module by: Phil Schniter. E-mail the author

Summary: Here we present some background material on multirate signal processing that is necessary to understand the filterbank processing used in sub-band coding. In particular, we describe modulation, upsampling, and downsampling in several domains: the time-domain, z-domain, and DTFT domain. In addition, we describe the aliasing phenomenon.

The presence of upsamplers and downsamplers in the diagram of Figure 2 from "Introduction and Motivation" implies that a basic knowledge of multirate signal processing is indispensible to an understanding of sub-band analysis/synthesis. This section provides the required background.

  • Modulation:
    Figure 1: Modulation using ejωonejωon
    This figure is a small flow chart. On the left is the variable x(n), with an arrow pointing to the right at a circle containing an x inside. Below the circle is the expression e^(jω_0n), and an arrow from this expression points up at the circle. To the right of the circle is an arrow pointing to the right at the variable y)(n).
    Figure 1 illustrates modulation using a complex exponential of frequency ωo. In the time domain,
    y(n)=x(n)ejωon.y(n)=x(n)ejωon.
    (1)
    In the z-domain,
    Y(z)=∑ny(n)z-n=∑nx(n)ejωonz-n=∑nx(n)e-jωoz-n=Xe-jωoz.Y(z)=∑ny(n)z-n=∑nx(n)ejωonz-n=∑nx(n)e-jωoz-n=Xe-jωoz.
    (2)
    We can evaluate the result of modulation in the frequency domain by substituting z=ejωz=ejω. This yields
    Y(ω)=∑ny(n)e-jωn=X(ω-ωo).Y(ω)=∑ny(n)e-jωn=X(ω-ωo).
    (3)
    Note that X(ω-ωo)X(ω-ωo) represents a shift of X(ω)X(ω)up by ωo radians, as in Figure 2.
    Figure 2: Frequency-domain effect of modulation by ejωonejωon.
    This figure contains two cartesian graphs, each plotting a horizontal axis ω and vertical axis X(ω) in the first and Y(ω) = X(ω - ω_0) in the second. In both graphs there is an identical triangle with one side along the horizontal axis. In the first graph, the triangle is centered so that its vertex that is not touching the horizontal axis is touching the vertical axis, leaving a portion of the triangle in quadrant II and a larger portion in quadrant I. In the second graph, the triangle is placed completely in the first quadrant, with one side still drawn along the horizontal axis and the leftmost vertex of the triangle touching the origin of the graph. The horizontal value of the location of the vertex that is not touching the horizontal axis is labeled as ω_0.
  • Upsampling:
    Figure 3: Upsampling by N.
    This is a small flowchart, beginning with the variable x(m), followed by an arrow pointing to the right at a circle labeled with an up arrow and the variable N, followed by another arrow pointing to the right at a final variable, y(n).
    Figure 3 illustrates upsampling by factor N. In words, upsampling means the insertion of N-1N-1 zeros between every sample of the input process. Formally, upsampling can be expressed in the time domain as
    y(n)=x(n/N)whenn=mNform∈Z0else.y(n)=x(n/N)whenn=mNform∈Z0else.
    (4)
    In the z-domain, upsampling causes
    Y(z)=∑ny(n)z-n=∑mx(m)z-mN=XzN,Y(z)=∑ny(n)z-n=∑mx(m)z-mN=XzN,
    (5)
    and in the frequency domain,
    Y(ω)=∑ny(n)e-jωn=XNω.Y(ω)=∑ny(n)e-jωn=XNω.
    (6)
    As shown in Figure 4, upsampling shrinks X(ω)X(ω) by a factor of N along the ω axis.
    Figure 4: Frequency-domain effects of upsampling by N=2N=2.
    This figure contains two cartesian graphs, each plotting a horizontal axis ω and vertical axis X(ω) in the first and Y(ω) = X(ω - ω_0) in the second. The first graph contains three identical triangles, each with one side sitting on the horizontal axis. The horizontal location of the triangles' vertices that are not located on the horizontal axis are labeled, with the leftmost triangles vertex at a horizontal value of -2π, the second with a value of 0, and the rightmost with a value of 2π. There are also ellipses to the left and right of this series of triangles, indicating that the pattern continues. The second graph is similar, except that the width of the base of the triangles is smaller. There are five pictured triangles, with the first, third, and fifth aligned in the same horizontal position as the three triangles in the first graph. The second and fourth triangles are placed evenly in between the aforementioned aligned triangles. This graph also includes ellipses on the left and right, indicating that the pattern continues.
  • Downsampling:
    Figure 5: Downsampling by N.
    This is a small flowchart, beginning with the variable x(n), followed by an arrow pointing to the right at a circle labeled with a down arrow and the variable N, followed by another arrow pointing to the right at a final variable, y(m).
    Figure 5 illustrates downsampling by factor N. In words, the process of downsampling keeps every NthNth sample and discards the rest. Formally, downsampling can be written as
    y(m)=x(mN).y(m)=x(mN).
    (7)
    In the z domain,
    Y(z)=∑my(m)z-m=∑mx(mN)z-m=∑nx˜(n)z-n/N,Y(z)=∑my(m)z-m=∑mx(mN)z-m=∑nx˜(n)z-n/N,
    (8)
    where
    x˜(n)=x(n)whenn=mNform∈Z0else.x˜(n)=x(n)whenn=mNform∈Z0else.
    (9)
    The neat trick
    1N∑p=0N-1ej2πNnp=1whenn=mNform∈Z0else1N∑p=0N-1ej2πNnp=1whenn=mNform∈Z0else
    (10)
    (which is not difficult to prove) allows us to rewrite x˜(n)x˜(n) in terms of x(n)x(n):
    Y(z)=∑nx(n)1N∑p=0N-1ej2πNnpz-n/N=1N∑p=0N-1∑nx(n)e-j2πNpz1/N-n=1N∑p=0N-1Xe-j2πNpz1/N.Y(z)=∑nx(n)1N∑p=0N-1ej2πNnpz-n/N=1N∑p=0N-1∑nx(n)e-j2πNpz1/N-n=1N∑p=0N-1Xe-j2πNpz1/N.
    (11)
    Translating to the frequency domain,
    Y(ω)=1N∑p=0N-1Xω-2πpN.Y(ω)=1N∑p=0N-1Xω-2πpN.
    (12)
    As shown in Figure 6, downsampling expands each 2π2π-periodic repetition of X(ω)X(ω) by a factor of N along the ω axis. Note the spectral overlap due to downsampling, called “aliasing.”
    Figure 6: Frequency-domain effects of downsampling by N=2N=2.
    This figure contains three cartesian graphs, each plotting a horizontal axis ω and vertical axis X(ω) in the first, X(ω/N) in the second, and Y(ω) in the third. The first graph contains five identical triangles, each with their base drawn on the horizontal axis. These triangles are evenly spaced, and the horizontal ω-value of the vertices that are not touching the horizontal axis are measured as -4π, -2π, 0 2π, and 4π. There are ellipses at the ends of this series, indicating that the pattern continues. In the second graph, there are three identical triangles with much wider bases than those in the first graph, although their width is not explicitly mentioned. The leftmost triangle is centered with top-vertex at horizontal value -4π. The second is centered at 0, and the third is at 4π. The third graph contains five triangle-shaped waves, although the troughs of the waves do not reach the horizontal axis. The peaks of the waves are at -4π, -2π, 0, 2π, and 4π.
  • Downsample-Upsample Cascade:
    Figure 7: N-Downsampler followed by N-upsampler.
    This is a small flowchart, beginning with the variable x(n), followed by an arrow pointing to the right at a circle labeled with an up arrow and the variable N, followed by another arrow pointing to the right at a circle containing a down arrow and the variable N, finally followed by an arrow pointing to the right at the expression y(n).
    Downsampling followed by upsampling (of equal factor N) is illustrated by Figure 7. This structure is useful in understanding analysis/synthesis filterbanks that lie at the heart of sub-band coding schemes. This operation is equivalent to zeroing all but the mNthmNth samples in the input sequence, i.e.,
    y(n)=x(n)whenn=mNform∈Z0else.y(n)=x(n)whenn=mNform∈Z0else.
    (13)
    Using trick Equation 10,
    Y(z)=∑ny(n)z-n=∑nx(n)1N∑p=0N-1ej2πNnpz-n=1N∑p=0N-1∑nx(n)e-j2πNpz-n=1N∑p=0N-1Xe-j2πNpz,Y(z)=∑ny(n)z-n=∑nx(n)1N∑p=0N-1ej2πNnpz-n=1N∑p=0N-1∑nx(n)e-j2πNpz-n=1N∑p=0N-1Xe-j2πNpz,
    (14)
    which implies
    Y(ω)=1N∑p=0N-1Xω-2πpN.Y(ω)=1N∑p=0N-1Xω-2πpN.
    (15)
    The downsampler-upsampler cascade causes the appearance of 2π/N2π/N-periodic copies of the baseband spectrum of X(ω)X(ω). As illustrated in Figure 8, aliasing may result.
    Figure 8: Frequency-domain effects of downsampler-upsampler cascade for N=2N=2.
    This figure contains two cartesian graphs, each plotting a horizontal axis ω and vertical axis X(ω) in the first and Y(ω) in the second. The first graph contains three identical triangles that are evenly spaced with one side sitting on the horizontal axis. The top vertex's horizontal position is measured and labeled as -2π, 0 and 2π from left to right. There are ellipses to the left and right of these triangles indicating that the pattern may continue horizontally in both directions. The second graph is a series of five connected triangular-shaped waves, where the peaks all reach the same height, and the troughs all to the same height, both above the horizontal axis. The graphs shows that the first peak occurs at a ω value of -2π, the third occurs at 0, and the fifth occurs at 2π. There are ellipses at the ends of these waves, indicating that the pattern may continue beyond the displayed portion of the graph.

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